J Theor Probab https://doi.org/10.1007/s10959-018-0819-z

Matrix Liberation Process I: Large Deviation Upper Bound and Almost Sure Convergence Yoshimichi Ueda1

Received: 12 April 2017 / Revised: 22 January 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018

Abstract We introduce the concept of matrix liberation process, a random matrix counterpart of the liberation process in free probability, and prove a large deviation upper bound for its empirical distribution and several properties on its rate function. As a simple consequence, we obtain the almost sure convergence of the empirical distribution of the matrix liberation process to that of the corresponding liberation process as continuous processes in the large N limit. Keywords Random matrix · Stochastic process · Unitary Brownian motion · Large deviation · Large N limit · Free probability Mathematics Subject Classification (2010) 60F10 · 15B52 · 46L54

1 Introduction Let M N (C)sa be all the N × N self-adjoint matrices endowed with the natural inner product A, BHS := Tr N (AB), and it has the following natural orthogonal basis:

This work was supported by Japan Society for the Promotion of Science (Grant-in-Aid for Challenging Exploratory Research JP16K13762).

B 1

Yoshimichi Ueda [email protected] Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan

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Cαβ

⎧ √1 (E αβ + E βα ) (1 ≤ α < β ≤ N ), ⎪ ⎪ ⎪ ⎨ 2 := E αα (1 ≤ α = β ≤ N ), ⎪ ⎪ ⎪ ⎩ √i (E αβ − E βα ) (1 ≤ β < α ≤ N ). 2

Here, Tr N stands for the non-normalized trace (i.e., Tr N (I N ) = N with the identity matrix I N ) and the E αβ are N × N standard matrix units. Using these inner product and orthogonal basis, we identify M N (C)sa with the N 2 -dimensional Euclidean space 2 R N , when we use usual stochastic analysis tools on Euclidean spaces. Choose the (i) n N 2 -dimensional standard Brownian motion Bαβ , 1 ≤ α, β ≤ N , 1 ≤ i ≤ n with natural filtration Ft , and define (i)

H N (t) :=

(i) N  Bαβ (t) √ Cαβ , t ≥ 0, 1 ≤ i ≤ n, N α,β=1

which are called the n independent N × N self-adjoint matrix Brownian motions on M N (C)sa . The stochastic differential equation (SDE in short) 1 (i) (i) (i) (i) (i) dU N (t) = i dH N (t) U N (t) − U N (t) dt with U N (0) = I N , 1 ≤ i ≤ n, 2 (i)

defines unique n independent diffusion processes U N , 1 ≤ i ≤ n, on the N × N unitary group U(N ), which are called the n independent N × N left unitary Brownian motions. It is known, see e.g., [13, Lemma 1.4(2)] and its proof, that they satisfy the so-called left increment property, that is, the U N(i) (t)U N(i) (s)∗ , t ≥ s, are independent of Fs and has the same distribution as that of U N(i) (t − s). This property plays a crucial role throughout this article. (i) For each 1 ≤ i ≤ n + 1, an r (i)-tuple i (N ) = (ξi j (N ))rj=1 of N × N selfadjoint matrices is given. Throughout this article, we assume that the given sequence n+1 are operator norm bounded, that is, ξi j (N ) ≤ R with some (N ) := (i (N ))i=1 constant R > 0, and has a limit joint distribution σ0 as N → ∞. See Sect. 2, item 3 for its precise formulation of σ0 . Here, we introduce the N × N matrix liberation process starting at (N ) as the multi-matrix-valued process   n+1  r (i) n+1 = (ξilib (N )(t)) t → lib (N )(t) = ilib (N )(t) j j=1 i=1 i=1 (i) (i) ∗ U N (t)ξi j (N )U N (t) (1 ≤ i ≤ n), with ξilib j (N )(t) := (i = n + 1). ξn+1 j (N ) We emphasize that the matrix liberation process lib (N ) is new in random matrix theory and also that each ilib (N ) is a constant process in distribution, that is, its empirical distribution is independent of time, but the whole family lib (N ) creates really non-commutative phenomena.

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The concept of matrix liberation process comes from the liberation process in free probability defined as follows. Let (M, τ ) be a tracial W ∗ -probability space, and Ai ⊂ M, 1 ≤ i ≤ n + 1, be unital ∗-subalgebras (possibly to be W ∗ subalgebras). Let vi , 1 ≤ i ≤ n, be n freely independent, left free unitary Brownian motions [2] in (M, τ ) with vi (0) = 1, which are (∗-)freely independent of the Ai . Then, the family consisting of Ai (t) := vi (t)Ai vi (t)∗ , 1 ≤ i ≤ n, and An+1 (t) := An+1 converges (in distribution or in moments) to a family of freely independent copies of Ai as t → ∞. Following Voiculescu [22], we call this ‘algebran+1 n+1 the liberation process starting at (Ai )i=1 . The valued process’ t → (Ai (t))i=1 lib matrix liberation process  (N ) is a natural random matrix model of the liberation process. The attempt of investigating the matrix liberation process lib (N ) is quite natural, because independent large random matrices are typical sources of free independence thanks to the celebrated work of Voiculescu [21] on the one hand and because, on the other hand, the concept of free independence is central in free probability theory and the liberation process is a ‘stochastic interpolation’ between a given statistical relation and the freely independent one in the free probability framework. The purpose of this article is to take a first step toward systematic study of the matrix (i) liberation process lib (N ) (rather than the unitary Brownian motions U N ) with the hope of providing a basis for the study of liberation process and free independence in view of random matrices. Here, we take a large deviation phenomenon for its empirical distribution, say τlib (N ) (see Sect. 2, item 2 for its formulation), as N → ∞, and actually prove a large deviation upper bound in scale 1/N 2 as N → ∞. The reader may think that a possible approach is to obtain a large deviation upper bound for the (i) U N at first and then to use the contraction principle. However, we do not employ such an approach, because we try to find the resulting formula of rate function in as direct a fashion as possible. In fact, the rate function that we will find is constructed by using a certain derivation that is similar to Voiculescu’s one in his liberation theory and shown to be good and to have a unique minimizer, which is identified with the empirical distribution σ0lib of the liberation process starting at the distribution σ0 (see Sect. 2, item 3 for its precise formulation). Hence, the standard Borel–Cantelli argument shows that τlib (N ) → σ0lib in the topology of weak convergence uniformly on finite time intervals almost surely as N → ∞. (See the end of the next section for several previously known related results.) Let us take a closer look at the contents of this article. The next section (Sect. 2) is concerned with the framework to capture empirical distributions τlib (N ) and σ0lib in terms of C ∗ -algebras. We emphasize that the C ∗ -algebra language is not avoidable if one wants to discuss the appropriate topology on the space of empirical distributions of non-commutative processes, because C ∗ -algebras are only appropriate, non-commutative counterparts of the spaces of continuous functions over topological spaces. Hence, Sect. 2 is just a collection of formulations for several concepts, but important to understand this article. We employ the strategy of the celebrated work on independent N × N self-adjoint Brownian motions due to Biane et al. [3] (also see [7, part VI, section 18]). Namely, we use the exponential martingale of the martingale

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  lib lib t → E tr N P ξ• (N )(·) | Ft − E tr N P ξ• (N )(·)

(1)

with tr N := N1 Tr N for any self-adjoint non-commutative polynomial P in indetermilib (N )(·)) denotes nates xi j (t), 1 ≤ i ≤ n + 1, 1 ≤ j ≤ r (i) and t ≥ 0, where P(ξ• (N )(t) for each x (t) into the polynomial P. Thus, we need the substitution of ξilib ij j to compute the resulting exponential martingale by giving the explicit formula of the quadratic variation of the martingale (1). This is done in Sect. 3 by utilizing the Clark– Ocone formula in Malliavin calculus. This is similar to [3], but we need some standard technology on SDEs in the framework of Malliavin calculus (e.g., [16, chapter 2]). The key of Sect. 3 is the introduction of a suitable non-commutative derivation, whose formula is not exactly same as but similar to the derivation in Voiculescu’s free mutual information [22]. This new derivation will further be investigated elsewhere. The resulting quadratic variation involves the conditional expectation with respect to the filtration Ft , and hence we need to investigate its large N limit in the time uniform fashion. This rather technical issue is the theme of Sect. 4, and the proof of the main result there is divided into two parts: We first describe the desired large N limit at each time, and then prove that the convergence is actually uniform in time. In the first part, we use the known convergence results on standard Gaussian self-adjoint random matrices, while in the second part the use of Thierry Lévy’s method [13] combining combinatorial techniques with the famous Itô formula is crucial. The rest of the discussion goes along a standard strategy in the large deviation theory for hydrodynamics. Namely, we need to prove the exponential tightness of the probability measures in question and introduce a suitable good rate function by looking at the quadratic variation computed in Sect. 3. These together with proving the large deviation upper bound are done in Sect. 5. In the same section, we give a few important properties on the rate function including the fact that σ0lib is its unique minimizer, and obtain the almost sure convergence of the empirical distribution τlib (N ) as continuous processes. The final section (Sect. 6) is a brief discussion on one of our ongoing works in this direction.

2 Empirical Distributions of (Matrix) Liberation Processes This section is devoted to a natural framework to capture the empirical distributions of (matrix) liberation processes.

Let C x• (·) := C {xi j (t)}1≤ j≤r (i),1≤i≤n+1,t≥0 be the universal unital ∗-algebra ∗ with subject to xi j (t) = xi j (t)∗ . We enlarge it to the  ∗universal enveloping C -algebra ∗ C R x• (·) with subject to xi j (t) ≤ R. Let T S C R x• (·) be all  the tracialstates on C ∗R x• (·). We denote by T S c C ∗R x• (·) the set of τ ∈ T S C ∗R x• (·) such that t → xiτj (t) := πτ (xi j (t)) ∈ πτ

  ∗ C R x• (·)  Hτ

define strong-operator continuous processes, where πτ : C ∗R x• (·) → B(Hτ ) denotes the GNS representation associated with τ and the natural lifting of τ to

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  πτ C ∗R x• (·) (the closure in the strong-operator topology) in Hτ is still denoted by the same symbol τ .   Lemma 2.1 For any τ ∈ T S C ∗R x• (·) , the following are equivalent:   (1) τ ∈ T S c C ∗R x• (·) . (2) For every ∈ N and any possible pairs (i 1 , j1 ), . . . , (i , j ), the function (t1 , . . . , t ) ∈ [0, +∞) → τ (xi1 j1 (t1 ) · · · xi j (t )) ∈ C is continuous. Proof (1) ⇒ (2) is trivial, since xi j (t) ≤ R. (2) ⇒ (1): For any monomial P = xi1 j1 (t1 ) · · · xi j (t ) one has, by assumption,  2  τ  (xi j (t) − xiτj (s)) τ (P) Hτ   = τ xi j (t ) · · · xi1 j1 (t1 )xi j (t)2 xi1 j1 (t1 ) · · · xi j (t )   − τ xi j (t ) · · · xi1 j1 (t1 )xi j (t)xi j (s)xi1 j1 (t1 ) · · · xi j (t )   − τ xi j (t ) · · · xi1 j1 (t1 )xi j (s)xi j (t)xi1 j1 (t1 ) · · · xi j (t )   + τ xi j (t ) · · · xi1 j1 (t1 )xi j (s)2 xi1 j1 (t1 ) · · · xi j (t ) → 0 (as t → s),

where τ : C ∗R xi j (·) → Hτ denotes the canonical map. Since xiτj (t) ≤  xi j (t) ≤ R as above, we conclude that t → xiτj (t) is strong-operator continuous.  Let W be the words of length in indeterminates xi j = xi∗j , 1 ≤ i ≤ n + 1, 1 ≤ of xik jk (t j ≤ r (i). For each w ∈ W , we denote by w(t1 , . . . , t ) the substitution   k) for xik jk into w = xi1 j1 · · · xi j . We introduce the function d : T S c C ∗R x• (·) × T S c C ∗R x• (·) → [0, +∞) by d(τ1 , τ2 ) ∞ ∞   :=

1

max

sup

2m (2R) w∈W (t1 ,...,t )∈[0,m] m=1 =1

  τ1 (w(t1 , . . . , t )) − τ2 (w(t1 , . . . , t ))

  for τ1 , τ2 ∈ T S c C ∗R x• (·) .     Lemma 2.2 (1) T S c C ∗R x• (·) , d is a complete metric space. (2) For any sequence (δk )k≥1 of positive real numbers, (δk )

⎫   ⎪     ∗  1⎬ c 2 1/2 τ ∈ T S C R x• (·)  sup := max τ (xi j (s) − xi j (t)) ≤  0≤s,t≤k 1≤ j≤r (i) ⎪ k⎪ ⎭ k≥1 ⎩  |s−t|≤δk 1≤i≤n+1 ⎧ ⎪ ⎨

  defines a compact subset in T S c C ∗R x• (·) endowed with d.

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  Proof (1) It is easy to see that d defines a metric on T S c C ∗R x• (·) . Thus, it suffices to confirm the completeness of the space.  Let τ p ∈ T S c C ∗R x• (·) be a Cauchy sequence, that is, d(τ p , τq ) → 0 as p, q → ∞. For every w = xi1 j1 · · · xi j ∈ W , we have |τ p (w(t1 , . . . , t )) − τq (w(t1 , . . . , t ))| ≤ 2m (2R) d(τ p , τq ) → 0 as p, q → ∞ for every (t1 , . . . , t ) ∈ [0, m] . Hence, lim p→∞ τ p (xi1 j1 (t1 ) · · · xi1 j (t )) exists for every word xi1 j1 (t1 ) · · · xi j (t ) in C x• (·). Since C x• (·) is the universal ∗-algebra generated by the xi j (t) = xi j (t)∗ , the words xi1 j1 (t1 ) · · · xi j (t ) together with the unit 1 form a linear basis. Hence, we can construct a linear functional τ on C x• (·) in such a way that τ (1) = 1 and τ (xi1 j1 (t1 ) · · · xi j (t )) = lim p→∞ τ p (xi1 j1 (t1 ) · · · xi j (t )); hence, τ (P) = lim p→∞ τ p (P) for every P ∈ C xi j (·) . Clearly, τ is a tracial state. We have |τ (P)| = lim p→∞ |τ p (P)| ≤ P for every P ∈ C x• (·) ( → C ∗R x• (·) naturally), and therefore, τ extends a tracial state on C ∗R x• (·). Fix w ∈ W and m ∈ N for a while. We have   τ p (w(t1 , . . . , t )) − τ (w(t1 , . . . , t ))   = lim τ p (w(t1 , . . . , t )) − τq (w(t1 , . . . , t )) q→∞

≤ 2m (2R) lim d(τ p , τq ) k →∞

for every (t1 , . . . , t ) ∈ [0, m] ; hence, sup (t1 ,...,t )∈[0,m]

  τ p (w(t1 , . . . , t )) − τ (w(t1 , . . . , t )) ≤ 2m (2R) lim d(τ p , τq ). q→∞

Thus, τ (w(t1 , . . . , t )) = lim p→∞ τ p (w(t1 , . . . , t )) is uniform in (t1 , . . . , t ) ∈ [0, m]  .∗ Since m ∈ N is arbitrary, we conclude, by Lemma 2.1, that τ ∈ c T S C R x• (·) . (2) Let τ p be an arbitrary sequence in (δk ) . For every m = 1, 2, . . . and every w ∈ W , the sequence of continuous functions τ p (w(t1 , . . . , t )) is equicontinuous on [0, m] , since  1/2    τ p (w(t1 , . . . , t )) − τ p (w(t  , . . . , t  )) ≤ R −1 τ p (xi j (tm ) − xi j (tm ))2 1 m=1

by the Cauchy–Schwarz inequality. Hence, for each m, = 1, 2, . . . , the Arzela– Ascoli theorem (see e.g., [18, Theorem 11.28]) guarantees that any subsequence of τ p has a subsequence τ p such that τ p (w(t1 , . . . , t )) converges uniformly on [0, m] as p  → ∞ for all w ∈ W (n.b. W is a finite set). Then, the usual diagonal argument with respect to = 1, 2, . . . enables us to select a subsequence τ p in such a way that for every w ∈ W , = 1, 2, . . . , the sequence of continuous functions τ p (w(t1 , . . . , t )) converges uniformly on [0, m] for as p  → ∞. This is done for each m and any given

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subsequence of τ p . Thus, by the usual diagonal argument again with respect to m, we can choose a common subsequence τ p that satisfies the same uniform convergence for all m. In the same way as in the  discussion about (1) above, we can construct a tracial state τ ∈ T S c C ∗R x• (·) in such a way that d(τ p , τ ) → 0 as p  → ∞. pair Moreover, for every   pair 0 ≤ s, t ≤ k with |s − t| ≤ δk and every possible (i, j), one has τ (xi j (s) − xi j (t))2 = lim p →∞ τ p (xi j (s) − xi j (t))2 ≤ 1/k 2 ,   and hence τ falls into (δk ) . We will provide some notations that will be used throughout the rest of this article.

1. Time-marginal tracial states: Let C ∗R x•  = C ∗R {xi j }1≤i≤n+1,1≤ j≤r (i) be the universal C ∗ -algebra generated by the xi j = xi∗j , 1 ≤ i ≤ n + 1, 1 ≤ j ≤ r (i) with subject to xi j  ≤ R. For each t := (t1 , . . . , tn+1 ) ∈ [0, +∞)n+1 , there exists a unique ∗-homomorphism (actually a ∗-isomorphism) πt : C ∗R x•  → C ∗R x• (·) sending xi j to xi j (ti ). When t := t1 = · · · = tn+1 , we simply  ∗ ∗ c write  ∗ πt : T S C R x• (·) →  π∗t := π t . The∗ πt induces a continuous map T S C R x•  by πt (τ ) := τ ◦ πt , where T S C R x•  is equipped with the w ∗ -topology. By Lemma 2.1, it is easy to see that t → πt∗ (τ ) is continuous for every τ ∈ T S c C ∗R x• (·) . We call πt∗ (τ ) the marginal tracial state of τ at multiple time t. 2. The empirical distribution τlib (N ) of lib (N ): The matrix liberation process   lib (N ) defines τlib (N ) ∈ T S c C ∗R x• ( · ) in such a way that    lib τlib (N ) (P) := tr N P ξ• (N )(·) ,

P ∈ C x• (·) .

We call this tracial state τlib (N ) the empirical distribution of the matrix liberation process lib (N ). The tracial state τlib (N ) is a random tracial state; actually, it (i)

depends upon the n independent left unitary Brownian motions U N via ξilib j (N ).  Hence, we have a Borel probability measure P(τlib (N ) ∈ · ) on T S c C ∗R x• (·) , and the large deviation upper bound that we will prove is about the sequence of probability measures P(τlib (N ) ∈ · ). 3. The empirical distribution σ0lib of the liberation process with initial distribution σ0 : The limit joint distribution σ0 of the sequence (N ) is defined to be a tracial state on C ∗R x•  naturally. Using its GNS construction and taking a suitable free product, we can construct self-adjoint random variables xiσj0 = xiσj0 ∗ , 1 ≤ i ≤ n + 1, 1 ≤ j ≤ r (i) and n freely independent, left free unitary Brownian motions vi , 1 ≤ i ≤ n, in a tracial W ∗ -probability space, say (L, σ˜ 0 ), in such a way that the joint distribution of the xiσj0 is indeed σ0 and that the xiσj0 and the vi are freely independent. Thanks to the universality of the C ∗ -algebra C ∗R x• (·), the strong-operator continuous processes ⎧ ⎨vi (t) x σ0 vi (t)∗ (1 ≤ i ≤ n), lib σ ij xi j0 (t) := ⎩x σ0 (i = n + 1) n+1 j   define a tracial state σ0lib ∈ T S c C ∗R x• (·) .

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Here is a simple fact.   Proposition 2.3 For every P ∈ C x• (·), we have lim N →∞ E τlib (N ) (P) =   σ0lib (P), that is, lim N →∞ E τlib (N ) ( · ) = σ0lib in the weak∗ -topology. Proof The proof of [2, Theorem 1(2)] works well without essential change.

 

This essentially known fact should be understood as a counterpart of the convergence of finite dimensional distributions, and will be strengthened to the convergence as continuous processes in Sect. 5.3. Namely, we will prove that the empirical distribution τlib (N ) itself converges to σ0lib in the metric d almost surely. Here, we briefly mention the known facts concerning the above proposition. The almost sure version (i.e., without taking the expectation E) of the above proposition has also been known so far (see e.g., the introduction of [5]); in fact, one can see it in the same way as in [2, Theorem 1(2)] with the use of more recent results, for example, [12, Proposition 6.9] and (the proof of) [9, Theorem 4.3.5] (see the comment just before Example 4.3.7 there). Moreover, its almost sure, strong convergence (i.e., the convergence of operator norms) version was recently established by Collins, Dahlqvist and Kemp [5]. In those results, the event of convergence (whose probability is of course 1) depends on the choice of time indices t1 , . . . , tk , unlike the fact that we will prove in Sect. 5.3.

3 Computation of Exponential Martingale It is easy to see that, as long as i = n + 1, 

ξilib j (N )(t), C αβ

 HS



= ξi j (N ), Cαβ HS    t  N  1 (i) i √ Cα  β  , ξilib (N )(s) , C dBα  β  (s) αβ j N HS α  ,β  =1 0  t  lib tr N (ξilib ds + j (N )(s))I N − ξi j (N )(s), C αβ

+

HS

0

(2) in the Euclidian coordinates on M N (C)sa with respect to the basis Cαβ . For a given P = P ∗ ∈ C x• (·), the matrix liberation process t → lib (N )(t) gives the (real-valued) bounded martingale M N in (1), that is,     M N (t) = E τlib (N ) (P) | Ft − E τlib (N ) (P) . The Clark–Ocone formula (see e.g., [10, Proposition 6.11] for any dimension and [16, subsection 1.3.4] for 1 dimension) asserts that M N (t) =

 n N   k=1 α  ,β  =1 0

123

t

 

    ,β  )  (k) lib P ξ E D(k;α tr (N )(·) Fs dBα  β  (s), N s •

J Theor Probab (k;α  ,β  )

(k)

where Ds denotes the Malliavin derivative in the Brownian motion Bα  β  explained in [16, p. 119]. The aim of this section is to compute this integrand explicitly by introducing a suitable non-commutative derivative. Observe that all the coefficients of SDE (2) are independent ofthe time parameter (k;α  ,β  ) lib ξi j (N )(t), Cαβ is well defined. and linear in the space variable. Thus, Ds (i)

HS

(i)

See e.g., [16, Theorem 2.2.1] for details. The function ξ → U N (t)ξU N (t)∗ is linear, (α,β) and hence the matrix-valued process Y (t) in [16, p. 126] is given by Y(α  ,β  ) (t) =   U N(i) (t)Cα  β  U N(i) (t)∗ , Cαβ . By (2), the formula [16, Eq. (2.59)] enables us to obtain HS that    ,β  ) lib ξ (N )(t), C D(k;α αβ s ij HS     (k) (k) (k) (k) U N (s)∗ Cα2 β2 U N (s), Cα1 β1 = δk,i 1[0,t] (s) U N (t)Cα1 β1 U N (t)∗ , Cαβ HS HS     1 × i √ Cα  β  , ξilib j (N )(s) , C α2 β2 N HS 

 δk,i 1[0,t] (s)  (k) (k) (k) (k) ∗ U N (t)U N (s)∗ i Cα  β  , ξilib = (N )(s) U (s)U (t) , C , √ αβ j N N HS N where we used the convention of summation over repeated indices (α1 , β1 ), (α2 , β2 ) as in [16, section 2.2]. For a while, we assume that P is a monomial in the ξilib j (N )(t). (k;α  ,β  )

By the Leibniz formula of Ds , we have, for any ζ ∈ C,   lib     Ds(k;α ,β ) Re tr N ζ P ξ• (N )(·) 1 = √ N 1 = √ N

     ,β  ) ξklibj (N )(t), Cαβ Re ζ tr N (Q 1 Cαβ Q 2 ) D(k;α s

N 



P=Q 1 xk j (t)Q 2 α,β=1 s≤t

 (k) (k) Re ζ tr N (Q 1 U N (t)U N (s)∗



P=Q 1 xk j (t)Q 2 s≤t

⎛ ⎜ ζi = Re ⎜ ⎝ √N

HS

 P=Q 1 xk j (t)Q 2 s≤t

  (k) (k) × i Cα  β  , ξklibj (N )(s) U N (s)U N (t)∗ Q 2 )  (k) tr N U N (s) ⎞ ⎟ 

(k) (k) (k) × ξk j (N ), U N (t)∗ Q 2 Q 1 U N (t) U N (s)∗ Cα  β  ⎟ ⎠,

lib (N )(·)) for short. Here and below, we used where we identify Q l , l = 1, 2, with % Q l (ξ• the convention that the summation P=Qxk j (t)R, s≤t above means that the resulting sum becomes 0 if no P = Qxk j (t)R with s ≤ t occurs. Therefore, we conclude that

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 n N  

 

     ,β  )  (k) lib ζ P ξ E D(k;α Re tr (N )(·) Fs dBα  β  (s) N s •

t

k=1 α  ,β  =1 0

=

n 



N 

⎛ t dB (k) α  β  (s)

√

k=1 α  ,β  =1 0

N

⎛

⎜ ⎜ ⎜ζ U (k) (s) tr × Re ⎜ N ⎝ N ⎝

⎡

⎤

⎢ ×E ⎢ ⎣i

 P=Q 1 xk j (t)Q 2 s≤t

⎞⎞

 ⎥ ⎟⎟  (k) (k) (k) ∗ ⎟⎟ ξk j (N ), U N (t)∗ Q 2 Q 1 U N (t) Fs ⎥ ⎦ U N (s) Cα  β  ⎠⎠ .

  Here, we have used the notation E[Y |Ft ] = E[Yi j |Ft ] for a matrix-valued random random variable Y = [Yi j ], where we naturally extend E[ − |Ft ]to complex-valued  variables. In the rest of this paper, we also write E[Y ] = E[Yi j ] . % We are now going back to a general P = P ∗ ∈ C x• (·). Write P = l ζl Pl with ζl ∈ C and monomials Pl in the ξilib j (N )(t). Then, we set  (k) (k) ζl U N (s) Z N (s) := ⎡ ⎢ × E⎢ ⎣i ⎡

l

 Pl =Q l1 xk j (t)Q l2 s≤t

⎢ = E⎢ ζl ⎣ l

⎤

 ⎥  (k) (k) (k) ∗ ξk j (N ), U N (t)∗ Q l2 Q l1 U N (t) Fs ⎥ ⎦ U N (s)



 ∗ (k) (k) U N (t)U N (s)∗

Pl =Q l1 xk j (t)Q l2 s≤t

⎤

  ⎥

 (k) (k) ×i ξklibj (N )(t), Q l2 Q l1 U N (t)U N (s)∗ Fs ⎥ ⎦,

which can be confirmed to be a self-adjoint matrix-valued random variable thanks to   lib (N )(·)) is real-valued, we have P = P ∗ . Since P = P ∗ , that is, tr N P(ξ•  t

n N          ,β  )  (k) lib M N (t) = E D(k;α Re tr N P ξ• (N )(·) Fs dBα  β  (s) s k=1 α  ,β  =1 0

=

 n N  

t

k=1 α  ,β  =1 0

=

 n N  

k=1 α  ,β  =1 0

123

t

   1 (k) (k) dBα  ,β  (s) √ Re tr N Z N (s) Cα  β  N   1 (k) (k) √ tr N Z N (s) Cα  β  dBα  ,β  (s) N

J Theor Probab

and the quadratic variation M N  of M N (t) becomes M N (t) =

 n N   k=1 α  ,β  =1 0

=

n  

k=1 0

=

t

n  t  k=1 0

t

 2 1 (k) tr N Z N (s)Cα  β  ds N

N 2 1   (k)   Z (s), C ds αβ N HS N3   α ,β =1

n    1  1  t  (k) 2  (k) 2 (s) ds = (s) ds,   Z Z N N HS tr N ,2 N3 N2 0 k=1

where we used a well-known formula on stochastic integrals (see e.g., [11, Proposition (k) (k  ) 3.2.17, Eq. (3.2.26)]) as well as Bαβ , Bα  β  (t) = δ(k,α,β),(k  ,α  ,β  ) t (see e.g., [11, Problem 2.5.5]). (k) Here, we introduce suitable non-commutative derivations to describe Z N (s). Definition 3.1 We expand C x• (·) into the universal ∗-algebra

C x• (·), v• (·) := C {xi j (t)}1≤ j≤r (i),1≤i≤n+1,t≥0  {vi (t)}1≤i≤n,t≥0 with subject to xi j (t) = xi j (t)∗ and vi (t)vi (t)∗ = 1 = vi (t)∗ vi (t), and define the derivations δs(k) : C x• (·) → C x• (·), v• (·) ⊗alg C x• (·), v• (·) by  δs(k) xi j (t) = δk,i 1[0,t] (s) xk j (t)vk (t − s) ⊗ vk (t − s)∗  − vk (t − s) ⊗ vk (t − s)∗ xk j (t)

for 1 ≤ k ≤ n. Let θ : C x• (·), v• (·) ⊗alg Cx• (·), v• (·) → C x• (·), v• (·) be a linear map defined by θ (Q ⊗ R) = R Q, and define (k) D(k) s := θ ◦ δs : C x • (·) → C x • (·), v• (·)

for 1 ≤ k ≤ n. (k) Although it is natural to define D(k) s to be −i θ ◦ δs , we drop the scalar multiple −i (k) in the definition for simplicity. It is easy to confirm that Z N (s) admits the following formula  

   (k) (•) (•) lib Z N (s) = E −i D(k) ξ• (N )(·), U N (· + s)U N (s)∗ Fs , s P

and hence we have the next proposition thanks to [11, Corollary 3.5.13].

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Proposition 3.2 For any P = P ∗ ∈ Cxi j (·), we have     M N (t) := E τlib (N ) (P) | Ft − E τlib (N ) (P)  t n N      tr N E −i D(k) = s P k=1 α  ,β  =1 0

     (•) (•) lib ξ• (N )(·), U N (· + s)U N (s)∗ Fs Cα  β 

×

(k)

dBα  β  (s) , √ N n   2

   1  t   (•) (•) lib ∗ M N  (t) = 2 P ξ (N )(·) , U (·+s)U (s) ds. E D(k) Fs  s • N N tr N ,2 N 0 ×

k=1

Therefore, Exp N (t)

,

,

:= exp N 1 − 2 n

2

    E τlib (N ) (P) | Ft − E τlib (N ) (P)

 t     2    (•) (•) lib ∗  P ξ (N )(·), U (· + s)U (s) Fs  E D(k) s • N N

k=1 0

tr N ,2

-ds

    becomes a martingale; hence, E Exp N (t) = E Exp N (0) = 1. For the later use, we remark that −i D(k) s P is self-adjoint (since so is P), and hence  2       (•) (•) lib ∗ P ξ (N )(·) , U (· + s)U (s) Fs  E D(k) s • N N tr N ,2 .   2 /   (•) (•) (k) lib ∗ . = −tr N E Ds P ξ• (N )(·) , U N (· + s)U N (s) Fs

(3)

4 Convergence of Conditional Expectation 4.1 Statement  ∗ s c For any given τ ∈ T S c (C ∗R x• (·))) and any s ≥ 0, we will construct   τ∗ ∈ T S  C R x• (·)) as follows. Taking a suitable free product, we expand πτ C R x• (·) , τ to a sufficiently larger tracial W ∗ -probability space, in which we can find n freely independent, left unitary free Brownian motions viτ , 1 ≤ i ≤ n, that are freely independent of the xiτj (t), 1 ≤ j ≤ r (i), 1 ≤ i ≤ n + 1, 0 ≤ t (≤ s if i = n + 1). Then, we define new strong-operator continuous processes

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J Theor Probab

τs

xi j (t) :=

⎧ ⎨v τ ((t − s) ∨ 0)x τ (t ∧ s)v τ ((t − s) ∨ 0)∗ (1 ≤ i ≤ n), i ij i ⎩x τ

n+1 j (t)

(i = n + 1).

It is known that there exists a unique τ -preserving conditional expectation E sτ onto the s von Neumann subalgebra generated by the xiτj (t) (= xiτj (t)), 1 ≤ i ≤ n, 1 ≤ j ≤ r (i),

τ τ 0 ≤ t ≤ s, and the xn+1 j (t) = x n+1 j (t), 1 ≤ j ≤ r (n + 1), t ≥ 0, in the ambient s

tracial W ∗ -probability space. Via the ∗-homomorphism sending xi j (t) to xiτj (t), we   obtain the desired tracial state τ s ∈ T S c C ∗R x• (·) . To each event E, we associate the essential supremum norm relative to E: s

X E := inf {L > 0 | P (E ∩ {|X | > L}) = 0} for every random variable X . Here is the main assertion of this section. Theorem 4.1 For any τ ∈ T S c (C ∗R x• (·))) and P1 , . . . , Pm ∈ C x• (·), v• (·), we have  

  (s)  lim lim sup tr N E P1,N  Fs ε0 N →∞ s≥0

   s  s 

 (s)  · · · E Pm,N  Fs − τ E sτ P1τ · · · E sτ Pmτ 

Oε (τ )

=0

with   (s) (•) (•) lib (N )(·), U N ((·) ∨ s)U N (s)∗ , Pk,N := Pk ξ•  s  s τ Pkτ := Pk x• (·), v•τ (( · − s) ∨ 0) 0   1 for 1 ≤ k ≤ m and with Oε (τ ) :=  d τlib (N ) , τ < ε , an event. Here, we use the  same convention such as E Pk,N  Fs as in Sect. 2. (k) By definition, Ds P with P ∈ C x• (·) is a linear combination of monomials of the form 1[0,t] (s) vk (t − s)∗ Q vk (t − s) with fixed Q ∈ C x• (·) and t ≥ 0. Hence, the next corollary immediately follows from Theorem 4.1.

Corollary 4.2 For any τ ∈ T S c (C ∗ x• (·)) and P ∈ C x• (·), we have  .    2 /   (•) (•) (k) lib ∗   Fs lim lim sup tr N E Ds P ξ• (N )(·), U N (· + s)U N (s) ε0 N →∞ s≥0  / .   s 2  τ τ  − τ E sτ D(k) P x (·), v (·) =0 s • •  Oε (τ )

for every 1 ≤ k ≤ n.

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4.2 Proof of Theorem 4.1 The proof is divided into two steps; we first prove in Sect. 4.2.1 that  



  (s)  (s)  Fs · · · E Pm,N Fs lim lim tr N E P1,N ε0 N →∞   s  s   − τ E sτ P1τ · · · E sτ Pmτ  Oε (τ )

=0 for each fixed s ≥ 0, and then in Sect. 4.2.2 that the convergence is actually uniform in time s. This strategy is motivated by Lévy’s work [13], and indeed his method is crucial in Sect. 4.2.2. A slight generalization of what Lévy established in [13] is necessary, and thus we will explain it in Sect. 4.3 for the reader’s convenience. Note that all the Pk is ‘supported’ in a finite time interval [0, T ], that is, the letters appearing in those Pk are from the xi j (t) and vi (t) with t ≤ T . Note also that we may and do assume that all the given Pk are monomials. 4.2.1 Convergence at Each Time s (i)

Choose another independent n-tuple VN of N × N left unitary Brownian motions (i) that are independent of the original n-tuple U N . Denote by EV the expectation only (i) in the stochastic processes VN . Define V ξilib j (N )s (t) :=

⎧ ⎨V (i) ((t − s) ∨ 0) ξ lib (N )(t ∧ s) V (i) ((t − s) ∨ 0)∗ (1 ≤ i ≤ n), N N ij ⎩ξ lib

n+1 j (N )(t)

= ξn+1 j (N )

(i = n + 1).

Then, it is not hard to see that 



 (s)  (•) lib E Pk,N  Fs = EV Pk ξ• (N )sV (·), VN (( · − s) ∨ 0) due to the left increment property of left unitary Brownian motions. V lib (N )V (·), V (•) (( · − s) ∨ 0)) depends only on a finite Note that Pk,s,N := Pk (ξ• s N (i)

(i)

number of VN (t) because we have fixed s. Each of those VN (t) is written as VN(i) (t) = W N(i,k) (t − (k/3))W N(i,k−1) (1/3) · · · W N(i,0) (1/3) or W N(i,0) (t) (i,l)

(i)

(i)

(i,l)

with W N (t) := VN (t + (l/3))VN (l/3)∗ , 0 ≤ t ≤ 1/3. Note that those W N (t) (0 ≤ t ≤ 1/3) become independent, N × N left unitary Brownian motions. In this V as a monomial in some ξilib way, we may think of Pk,s,N j (N )(t) (with t ≤ s as long (i,l)

(i,l)

as i = n + 1) and some W N (t), W N (t)∗ with 0 ≤ t ≤ 1/3. Accordingly, we τ (t) := v τ (t + (l/3))v τ (l/3)∗ , 0 ≤ t ≤ 1/3, l ∈ N, which become left write wi,l i i

123

J Theor Probab τ (·), v τ (( · − s) ∨ 0)) is also the free unitary Brownian motions. Then, Pkτ = Pk (x• • V τ τ (t) for ξ lib (N )(t) and same monomial as Pk,s,N with the substitution of xi j (t) and wi,l ij s

s

(i,l)

W N (t), respectively. Consequently, it suffices, for the purpose here, to prove that         lim lim tr N EW Q 1,N · · · EW Q m,N − τ E sτ (Q τ1 ) · · · E sτ (Q τm ) O

ε (τ )

ε0 N →∞

=0 (4)

with   (•,◦) lib (N )(·), W N (·) , Q k,N := Q k ξ•

 τ  τ Q τk := Q k x• (·), w•,◦ (·) , 1 ≤ k ≤ m

for any given monomials Q 1 , . . . , Q m in indeterminates xi j (t) (with 0 ≤ t ≤ s as long as i = n + 1), wi,l (t), wi,l (t)∗ with 0 < t ≤ 1/3, where EW denotes the expectation (i,l) only in the stochastic processes W N and also Q k,N and Q τk are defined similarly as above. Note that the given monomials Q 1 , . . . , Q m depend only on a finite number of indeterminates xi1 j1 (t1 ), . . . , xi p j p (t p ), xn+1 j p+1 (t p+1 ), . . . , xn+1 j p (t p ) (with 1 ≤ i 1 , . . . , i p ≤ n, 0 ≤ t1 , . . . , t p ≤ s) and wi1 l1 (t1 ), . . . , wiq lq (tq ) (with 0 < t1 , . . . , tq ≤ 1/3). As in [5, section 4], we may and do write wiτk lk (tk ) = f tk (gik lk ), where f tk is a continuous function from the real line R to the onedimensional torus T (depending only on the time tk ) and a standard semicircular system gi1 l1 , . . . , giq lq , which is freely independent of xiτ1 j1 (t1 ), . . . , xiτp j p (t p ) τ τ and xn+1 j p+1 (t p+1 ), . . . , x n+1 j p (t p ). Accordingly, by [5, Proposition 4.3] we can choose an independent family of N × N standard Gaussian self-adjoint random (i ,l ) (i ,l ) matrices G N1 1 , . . . , G Nq q in such a way that they are independent of the (i )

(i )

U N 1 (t1 ), . . . , U N p (t p ) (corresponding to indeterminates xi1 j1 (t1 ), . . . , xi p j p (t p )) and the operator norm W N(ik ,lk ) (tk ) − f tk (G (iNk ,lk ) ) M N (C) → 0 almost surely as N → ∞. For any x, y ∈ C N with xC N ≤ 1, yC N ≤ 1, we have

 



        (•,◦) lib (N )(·), W (•,◦) (·) − E lib xy N  EW Q k ξ• G Q k ξ• (N )(·), f (·) G N (·) N 

      C    (•,◦) (•,◦) lib lib =  EW ∪G ξ• (N )(·), W N (·)) − Q k ξ• (N )(·), f (·) G N (·) xy N  C

          (•,◦) lib (N )(·), W (•,◦) (·) − Q ξ lib (N )(·), f ≤ EW ∪G  Q k ξ• G (·) xy N k (·) • N N   C       (•,◦) (•,◦) lib (N )(·), W lib (N )(·), f ≤ EW ∪G Q k ξ• (·) − Q k ξ• (·) G N (·)  N

     

M N (C)

(i,l)

with the expectations EG and EW ∪G only in the variables G N (i,l) G N , respectively. Hence, we conclude that

(i,l)

and the W N (t),

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      lib lib (N )(·), W N(•,◦) (·) − EG Q k ξ• (N )(·), f (·) G (•,◦)  EW Q k ξ• N (·) M N (C)          (•,◦) (•,◦) lib lib ≤ EW ∪G  Q k ξ• (N )(·), W N (·) − Q k ξ• (N )(·), f (·) G N (·)  M N (C)       (i ,l ) (i ,l )  , ≤ Const. E max W N k k (tk ) − f tk G Nk k  M N (C)

1≤k≤q

    since ξilib j (N )(t)

(i,l)

M N (C)

(i,l)

≤ R and since all the W N (t) and f t (G N (t)) are unitary (i ,l )

(i ,l )

matrices. Since max1≤k≤q W N k k (tk ) − f tk (G Nk k ) M N (C) → 0 almost surely as N → ∞, we conclude that       lim tr N EW Q 1,N · · · EW Q m,N N →∞ 



 (5) ( f (G )) ( f ∗ (G ∗ ))  − tr N EW ∪G Q 1,N∗ ∗ · · · EW ∪G Q m,N  =0 ∞

with ( f (G ∗ ))

∗ Q k,N

   (•,◦) lib := Q k ξ• (N )(·), f (·) G N (·) ,

where  − ∞ denotes the essential supremum norm. For a given 0 < δ ≤ 1, the Weierstrass theorem enables us to choose a polynomial ptk so that the supremum norm  ptk − f tk [−3,3] over the interval [−3, 3] is not greater than δ. For a while, we fix such polynomials ptk , 1 ≤ k ≤ q. Since wiτk lk (tk ) = f tk (gik lk ) and gik lk  ≤ 2, it immediately follows that there exists a positive constant C > 0 such that            (τ, p (g )) (τ, p (g ))  ≤ Cδ (6) τ E sτ (Q τ1 ) · · · E sτ (Q τm ) − τ E sτ Q 1 ∗ ∗ · · · E sτ Q m ∗ ∗ with (τ, p∗ (g∗ ))

Qk

 τ  := Q k x• (·), p(·) (g•◦ ) , 1 ≤ k ≤ m.

Consider the event E N :=

q 2    (ik ,lk )  G N  k=1

M N (C)

3 3 ,

whose probability P(E N ) is known to converge to 1 as N → ∞ (see e.g., [1, subsection 5.5] and references therein). Similarly as above, we can find a universal constant C  > 0 so that  



  ( f (G )) ( f ∗ (G ∗ )) tr N EG 1E N Q 1,N∗ ∗ · · · EG 1E N Q m,N 



  (7)  ( p (G )) ( p∗ (G ∗ )) − tr N EG 1E N Q 1,N∗ ∗ · · · EG 1E N Q m,N  ≤ C  δ,

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with ( p (G ∗ ))

Q k,N∗

   (•,◦) lib := Q k ξ• (N )(·), p(·) G N (·) ,

   (i ,l ) (i ,l )  since  f tk (G Nk k ) − ptk (G Nk k )

M N (C)

≤ δ on the event E N . By the ‘Cauchy–

Schwarz inequality’ for matricial expectations (see Remark 4.5 below), we have 

  ( p (G ))  EG 1\E N Q k,N∗ ∗ 

M N (C)

  1/2   1/2   ( p (G )) ∗ ( p (G ))  ≤ EG 1\E N I N  M (C) EG Q k,N∗ ∗ Q k,N∗ ∗  N M N (C)    1/2 ( p∗ (G ∗ )) ∗ ( p∗ (G ∗ ))  1/2  = (1 − P(E N )) EG Q k,N Q k,N 

(8)

M N (C)

and similarly 

  ( f ∗ (G ∗ ))   EG 1\E N Q k,N M N (C) 

  ( f ∗ (G ∗ )) ∗ ( f ∗ (G ∗ )) 1/2 ≤ (1 − P(E N ))1/2 EG (Q k,N ) Q k,N  M N (C)  1/2   ( f∗ (G ∗ )) ∗ ( f∗ (G ∗ ))  ≤ (1 − P(E N ))1/2 EG (Q k,N ) Q k,N 

(9)

M N (C)

≤

Ck (1 − P(E N ))1/2 (i ,l )

with some constant Ck > 0 depending only on Q k , since the f tk (G Nk k ) are unitary matrices and ξilib j (N )(t) M N (C) ≤ R. Since P(E N ) → 1 as N → ∞ as remarked before, we need to prove that      ( p (G )) ∗ ( p (G ))  sup max EG Q k,N∗ ∗ Q k,N∗ ∗ 

N ∈N 1≤k≤m

M N (C)

< +∞

(10)

and   



  ( p (G )) ( p∗ (G ∗ )) lim lim tr N tr N EG Q 1,N∗ ∗ · · · EG Q m,N ε0 N →∞      (τ, p (g )) (τ, p (g ))  −τ E sτ Q 1 ∗ ∗ · · · E sτ Q m ∗ ∗ 

Oε (τ )

= 0,

(11)

both of which are similar to what Biane et al. proved in [3, section 4]. However, we will give more ‘exact’ proofs to them later for the sake of completeness. In fact, (8) and (10) imply that 

    ( p∗ (G ∗ ))   1 Q lim  E  G \E N k,N  N →∞ M N (C) 

∞

= 0, 1 ≤ k ≤ m,

(12)

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and moreover, by (9) 

  ( f ∗ (G ∗ ))  lim   EG 1\E N Q k,N N →∞

   M N (C) 

∞

= 0, 1 ≤ k ≤ m.

(13)

    (i ,l ) Remark that 1E N ptk (G Nk k ) M N (C)  ≤  ptk [−3,3] ≤ 1 + δ ≤ 2. By (5)–(7) ∞ and (11)–(13), we have         lim lim tr N EW Q 1,N · · · EW Q m,N − τ E sτ (Q τ1 ) · · · E sτ (Q τm ) O

ε (τ )

ε0 N →∞ 

≤ (C + C)δ.

Hence, (4) follows because δ > 0 can arbitrarily be small and both C and C  are independent of the choice of δ > 0. Hence, we have completed the first step expect showing (10) and (11).   Here, we prove (10) and (11). We need two simple lemmas, which are of independent interest because they are very explicit. Lemma 4.3 Let G (i) be an independent sequence of N × N standard Gaussian selfadjoint random matrices, and A(i) be an sequence of N × N deterministic matrices. Then, we have 

E G ( p(1)) A(q(1)) · · · A(q( −1)) G ( p( ))

  = N (γ π )−1− /2 tr π γ A(q(1)) , . . . , A(q( −1)) , ∗ . p

π ∈P2 ( ) p

Here, P2 ( ) is the set of all permutations π with p◦π = p whose cycle decompositions consist only of transpositions, γ denotes (1, 2, . . . , ), (π γ ) is the number of cycles  in π γ , and finally tr σ A(q(1)) , . . . , A(q( −1)) , ∗ , σ ∈ S , is defined as follows. If σ is a cycle (i 1 , . . . , i k ), then it becomes ⎧ (q(i )) (q(i )) ⎪ (i 1 , . . . , i k ≤ − 1), ⎪tr N (A 1 · · · A k )I N ⎨ A ⎪ ⎪ ⎩I

(q(i j+1 ))

N

· · · A(q(ik )) A(q(i1 )) · · · A(q(i j−1 )) (i j = ), (k = 1, i 1 = ),

and generally it is to be



  tr σ1 A(q(1)) , . . . , A(q( −1)) , ∗ · · · tr σm A(q(1)) , . . . , A(q( −1)) , ∗ with cycle decomposition σ = σ1 · · · σm (n.b., only one cycle σk contains ; hence, no ambiguity occurs in the above product because its factors commute with each other). Note that tr N (tr σ [A1 , . . . , A −1 , ∗ ] A ) = tr σ [A1 , . . . , A −1 , A ] with the notation of [15, Proposition 22.32] on the right-hand side. This is the key of the proof below.

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    Proof Remark that tr N (X A) = tr N E G ( p(1)) A(q(1)) · · · A(q( −1)) G ( p( )) A for all   ( p(1)) (q(1)) A · · · A(q( −1)) G ( p( )) . This together with [15, PropoA forces X = E G sition 22.32] (see the above remark) implies the desired result.   This lemma immediately implies (10), because ξilib j (N )(t) M N (C) ≤ R and (γ k π ) − 1 − k /2 ≤ 0, see e.g., [15, Exercise 22.15]. Similarly as above, we derive the next lemma from [15, Proposition 22.33] and its discussion there. Lemma 4.4 Let (M, τ ) be a tracial W ∗ -probability space. Let gi be a freely independent sequence of standard semicircular elements in (M, τ ), and ai be a sequence of elements in M which are freely independent of the gi . Let E be a unique τ -preserving conditional expectation onto the von Neumann subalgebra generated by the ai . Then, we have      E g p(1) aq(1) · · · aq( −1) g p( ) = τπ γ aq(1) , . . . , aq( −1) , ∗ , p

π ∈N C2 ( ) p

p

where N C2 ( ) is the subset of all π ∈ P2 ( ) that are non-crossing as partitions. The other undefined symbol τπ γ [aq(1) , . . . , aq( −1) , ∗ ] is similarly defined as in the previous lemma. It is not so hard to derive (11) from Lemmas 4.3, 4.4 in the following way: For simplicity, we write ( p (G ∗ ))

Q k,N∗

(τ, p (g )) Qk ∗ ∗ ( p(·))

where each G k

(q(0))

= Ak =

( p(1))

Gk

(q(1))

Ak

(q( k −1))

· · · Ak

( p( k ))

Gk

(q( k ))

Ak

,

(k) (k) (k) (k) aq(0) g p(1) aq(1) · · · aq( g (k) a (k) , k −1) p( k ) q( k ) (q(·))

(or Ak

(i,l)

) is some of the G N

(resp. a product of some ξilib j (N )(t) (k)

(k)

(t ≤ s as long as i = n + 1) or I N ) and accordingly, each g p(·) (or aq(·) ) is some of the gil (resp. a product of some xiτj (t) (t ≤ s as long as i = n + 1) or 1). Remark that (γ k π )−1− k /2 is always non-positive and equals 0 if and only if π is non-crossing, see e.g., [15, Exercise 22.15]. Hence, by Lemmas 4.3 and 4.4 we have

 ( p (G )) EG Q k,N∗ ∗ =



(q(0))

Ak

 (q(1)) (q( −1)) (q( )) tr π γ k Ak , . . . , Ak k , ∗ Ak k

p

π ∈N C2 ( k )



+ p

N (γ k π )−1− k /2 p

π ∈P2 ( k )\N C2 ( k ) (q(0))

  (τ, p (g )) E sτ Q k ∗ ∗ =

× Ak  p

 (q(1)) (q( −1)) (q( )) tr π γ k Ak , . . . , Ak k , ∗ Ak k ,

 (k) (k) (k) (k) aq(0) tr π γ k aq(1) , . . . , aq( k −1) , ∗ aq( k ) .

π ∈N C2 ( k )

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Therefore, by ξilib j (N )(t) M N (C) ≤ R, we obtain that  



  ( p (G )) ( p∗ (G ∗ )) tr N EG Q 1,N∗ ∗ · · · EG Q m,N        (τ, p (g )) (τ, p (g )) − τ E sτ Q 1 ∗ ∗ · · · E sτ Q m ∗ ∗    C  ≤  S(τlib (N ) (W1 ), . . . , τlib (N ) (W L )) − S(τ (W1 ), . . . , τ (W L )) + N with some monomials W1 , . . . , W L in the xi j (t) and a positive constant C  > 0 (which is independent of N ), where S is a certain polynomial of commutative indeterminates. It follows that  



  ( p (G )) ( p∗ (G ∗ )) lim tr N EG Q 1,N∗ ∗ · · · EG Q m,N N →∞       (τ, p (g )) (τ, p (g )) − τ E sτ Q 1 ∗ ∗ · · · E sτ Q m ∗ ∗  Oε (τ )   ≤  S(τlib (N ) (W1 ), . . . , τlib (N ) (W L )) − S(τ (W1 ), . . . , τ (W L ))O (τ ) . ε

By definition, for a given δ > 0, there exists ε > 0 so that for every 0 < ε ≤ ε one has    S(τlib (N ) (W1 ), . . . , τlib (N ) (W L )) − S(τ (W1 ), . . . , τ (W L ))

Oε (τ )

≤ δ.

Hence, we are done.

 

Remark 4.5 (See e.g., [17, Exercise 3.4 in p. 40]) Let X = [X i j ] be a matrix whose entries are integrable. If X is positive definite  almost surely, then so is E[X ] = [E[X i j ]] % % ¯ ¯ since i, j ζi E(X i j )ζ j = E i, j ζi X i j ζ j ≥ 0 for any scalars ζi . Let A = [Ai j ] and B = [Bi j ] be N × N matrices whose entries have all moments. Since     ∗  A B E[A∗ A] E[A∗ B] A B = E ≥ O, E[B ∗ A] E[B ∗ B] O O O O one has, for all t, θ ∈ R and x, y ∈ C N ,   t 2 (E[A∗ A]x|x)C N + 2t Re e−iθ (E[B ∗ A]x|y)C N + (E[B ∗ B]y|y)C N  /  . tx  tx E[A∗ A] E[A∗ B] = ≥ 0, E[B ∗ A] E[B ∗ B] eiθ y  eiθ y C2N  2 and hence (E[B ∗ A]x|y)C N  ≤ (E[A∗ A]x|x)C N (E[B ∗ B]y|y)C N . It follows that E[B ∗ A]2M N (C) ≤ E[A∗ A] M N (C) E[B ∗ B] M N (C) .

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J Theor Probab

4.2.2 The Convergence is Uniform in Time s Let us introduce the map s : C x• (·), v• (·) → C x• (·), v• (·) defined by replacing xi j (t) with vi ((t − s) ∨ 0)xi j (t ∧ s)vi ((t − s) ∨ 0)∗ as long as i = n + 1 and also replacing vi (t) with vi ((t − s) ∨ 0), with keeping the other letters. Remark that the resulting s P is a (non-commutative) polynomial in the xi j (t) (with t ≤ s if i = n + 1) and the vi (t). For a while, we are dealing with an arbitrarily fixed monomial P whose letters are supported in [0, T ], that is, the letters are from the xi j (t) and the vi (t) with t ≤ T , and so is s P. As before we have  

    (•) (•) (•) lib lib E P ξ• (N )(·), U N ((·) ∨ s)U N (s)∗ Fs = EV (s P) ξ• (N )(·), VN (·) , where VN(i) , 1 ≤ i ≤ n, are n independent left unitary Brownian motions that are independent of the U N(i) (t) with t ≤ s. Denote by L(P) the number of letters in the given monomial P (we call it the length of P). Observe that L(s P) ≤ 3L(P). In what follows, we fix s, but will give our desired estimate in such a way that it is independent of the choice of s. Let us introduce the following %n algorithm: If vi (t1 ), vi (t2 ), . . . , vi (t i ) with t1 < i ≤ L(s P) ≤ 3L(P)) are all the vi (·) letters t2 < · · · < t i (n.b., i ≤ i=1 appearing in s P, we replace these with wi1 , wi2 wi1 , wi3 wi2 wi1 , . . . , wi i · · · wi2 wi1 with new indeterminates wi j (1 ≤ i ≤ n, 1 ≤ j ≤ i (≤ 3L(P))), and set ti j := t j − t j−1 with t0 := 0. Applying this algorithm to the monomial s P, we get a new 4 s P whose letters are in the xi j (t) (0 ≤ t ≤ s) and the wi j . Observe the monomial  following rather rough estimates 4 s P) ≤ L(s P)2 ≤ 9L(P)2 , L(

ti j ≤ T.

(14)

(i, j)

Let W N be independent left unitary Brownian motions that are independent of the (i) U N (t) with t ≤ s and denote by EW the expectation only in the stochastic processes (i, j) W N . By the left increment property of left unitary Brownian motions, we have  

  lib (N )(·), U N(•) ((·) ∨ s)U N(•) (s)∗ Fs E P ξ•

  (•) lib = EV (s P) ξ• (N )(·), VN (·)

  (•,◦) lib 4 s P) ξ• = E W ( (N )(·), W N (t•◦ ) , lib (N )(·), W (•,◦) (t )) denotes the substitution of ξ lib (t) and W (i, j) (t ) 4 s P)(ξ• where ( •◦ ij N ij N 4 for xi j (t) and wi j , respectively, into s P.

123

J Theor Probab

4 s P, and write X = X (1) · · · X ( ) with := For simplicity, let us denote X :=  L(X ) whose letters X (k) are from {xi j (t) | 1 ≤ i ≤ n, 1 ≤ j ≤ r (i), 0 ≤ t (≤ s as long as i = n + 1)} ∪ {wi j , wi∗j | 1 ≤ i ≤ n, 1 ≤ j ≤ 3L(P)}. The substitution (i, j)

of ξilib j (N )(t) and W N (ti j ) for x i j (t) and wi j , respectively, into the monomial X is denoted by X N = X N (1) · · · X N ( ). Let X N ( + 1) := A ∈ M N (C) be arbitrarily chosen. Let ρ : C[Sl+1 ]  (C N )⊗( +1) , on which M N (C)⊗( +1) acts naturally, be the permutation representation of S +1 over the tensor product components; in fact, ρ(σ )(e1 ⊗ · · · ⊗ e +1 ) = eσ −1 (1) ⊗ · · · ⊗ eσ −1 ( +1) for σ ∈ S +1 . For each σ ∈ S +1 we define, following [13, section 3] (rather than Lemma 4.3), tr σ (X N (1), . . . , X N ( ), X N ( + 1)) 1 ⊗( +1) := (σ ) Tr N (ρ(σ ) (X N (1) ⊗ · · · X N ( ) ⊗ X N ( + 1))) N 5 tr N (X N (k∗ ) · · · X N (k1 )), = (k1 ,...,k∗ )σ

where (k1 , . . . , k∗ )  σ means that (k1 , . . . , k∗ ) is a cycle component of the cycle decomposition of σ ∈ S +1 , and (σ ) denotes the number of cycles in σ as in the previous subsection. Note that tr σ (· · · ) here is not consistent with tr σ [· · · ] in Lemma 4.3, but tr σ −1 (· · · ) = tr σ [· · · ] holds. In particular, for the cycle γ +1 = (1, . . . , , + 1) we have

 EW tr γ −1 (X N (1), . . . , X N ( ), A) = tr N (EW [X N ] A) . +1

Then, by a slight generalization of [13, Proposition 3.5] (see the next subsection for its precise statement with a detailed proof) there exist universal coefficients cσ , σ ∈ S +1 , depending on the ti j and X , and a universal constant C > 1, depending only on T and L(P) due to (14) (and hence only on P), such that        C (0) (0) tr N (EW [X N ] A) − cσ tr σ X N (1), . . . , X N ( ), A  ≤ 2 Atr N ,1  N   σ ∈S +1 and   cσ  ≤ C, σ ∈ S +1 with (0) X N (k)

123

:=

IN

(X (k) = wi j or wi∗j ),

X N (k) (otherwise),

(15)

J Theor Probab

   (0)  ≤ (R ∨ 1) L(P) A1,tr (n.b., the procedure since tr σ X (0) N N (1), . . . , X N ( ), A 4 s P does not make the number of xi j (t) increase), where  − 1,tr N from P to X [P] =  denotes the trace norm with respect to the normalized trace tr N . Since   (0) (0) cσ tr σ X N (1), . . . , X N ( ), A

 σ ∈S +1

= tr N

. 

 / (0) (0) cσ tr σ −1 X N (1), . . . , X N ( ), ∗ A

σ ∈S +1

with the notation in Lemma 4.3 and since A ∈ M N (C) is arbitrary, we conclude that   

    (0) (0) EW [X N ] − cσ tr σ −1 X N (1), . . . , X N ( ), ∗      σ ∈S +1

≤ M N (C)

C . N2

(16)

 (0) (0) Notice that tr σ −1 X N (1), . . . , X N ( ), ∗ depends only on the traces tr N of monomials in the ξilib j (N )(t) (with 0 ≤ t ≤ s as long as i  = n + 1), or other words, the τlib (N ) of monomials in the xi j (t) (with 0 ≤ t ≤ s as long as i = n + 1). Observe that (16) holds for any monomial P and s ≥ 0, and we should write 4 s P, = P (= L(X [P])), cσ = cσ (P) and C = C P for clarifying X = X [P] :=  the dependency in what follows. Set X [P]

(0)

(k) :=

1

(X [P](k) = wi j or wi∗j ),

X [P](k) (otherwise)

and for simplicity we write

 (0) (0) τlib (N ) (σ ; P) := tr σ −1 X [P] N (1), . . . , X [P] N ( P ), ∗ ,

 τ (σ ; P) := τσ −1 X [P](0) (1), . . . , X [P](0) ( P ), ∗ ,  E(P; τlib (N ) ) := cσ (P) τlib (N ) (σ ; P), σ ∈S P +1

E(P; τ ) :=



cσ (P) τ (σ ; P).

σ ∈S P +1

We are now finalizing the proof by using what we have prepared so far. Let P1 , . . . , Pm be any monomials such as the above P, that is, all the letters appearing in those are supported in a finite time interval [0, T ], and rewrite k := Pk = L(X [Pk ]) and set L := max{L(Pk ) | 1 ≤ k ≤ m} and C0 := max{C Pk | 1 ≤ k ≤ m} for simplicity. We have

123

J Theor Probab

      tr N E(P1 ; τlib (N ) ) · · · E(Pm ; τlib (N ) ) − τ (E(P1 ; τ ) · · · E(Pm ; τ ))    

≤

  cσ (P1 ) · · · cσ (Pm ) m 1

σk ∈S k +1 (1≤k≤m)

      × tr N τlib (N ) (σ1 ; P1 ) · · · τlib (N ) (σm ; Pm ) − τ (τ (σ1 ; P1 ) · · · τ (σm ; Pm ))  ≤



C0m

σk ∈S k +1 (1≤k≤m)

    tr N τlib (N ) (σ1 ; P1 ) · · · τlib (N ) (σm ; Pm ) 

  − τ (τ (σ1 ; P1 ) · · · τ (σm ; Pm ))  (17) by (15). Let σk = be the cycle decomposition such that the rightmost ((σ )) cycle σk k contains k + 1. Then, we may and do write ((σ )) σk(1) · · · σk k

  ((σ )−1) ((σ )) (1) Q k,N k , τlib (N ) (σk ; Pk ) = τlib (N ) (Q k ) · · · τlib (N ) Q k k   ((σk )−1) ((σ )) Qk k τ (σk ; Pk ) = τ (Q (1) k ) · · · τ Qk (∗)

with some monomials Q k in the xi j (t), i = n + 1, 0 ≤ t ≤ T , and the xn+1 j (t), ((σ )) whose total length is at most L(Pk ) ≤ L by construction, possibly with Q k k = 1, ((σ )) ((σk )) . It follows where Q k,N k denotes the substitution of ξilib j (N )(t) for x i j (t) into Q k that       tr N τlib (N ) (σ1 ; P1 ) · · · τlib (N ) (σm ; Pm ) − τ (τ (σ1 ; P1 ) · · · τ (σm ; Pm ))    ,  m    5 ((σ )−1) ((σ )) (1) m )) ≤  τlib (N ) (Q k ) · · · τlib (N ) (Q k k ) τlib (N ) Q 1 1 · · · Q ((σ m k=1 m 5

, − ,

((σk )−1) τ (Q (1) ) k ) · · · τ (Q k

   ((σ )) m ))  τ Q 1 1 · · · Q ((σ m 

k=1

m    ≤ 1+ ((σk ) − 1) × (R ∨ 1) Lm × 2T +1 (2(R ∨ 1)) Lm d τlib (N ) , τ k=1

  ≤ (9L m + 1) · (R ∨ 1) Lm · 2T +1 (2(R ∨ 1)) Lm d τlib (N ) , τ . 2

(18) Remark that EW [X [Pk ] N ] M N (C) ≤ X [Pk ] N  M N (C) ≤ (R ∨ 1) L by construction, since the matricial expectation EW [ − ] is a unital positive map, see Remark 4.5. Therefore, (16)–(18) altogether imply that

123

J Theor Probab

    tr N (EW [X [P1 ] N ] · · · EW [X [Pm ] N ]) − τ (E(P1 ; τ ) · · · E(Pm ; τ ))    ≤

  C1 + C2 d τlib (N ) , τ 2 N

with constants C1 , C2 > 0 that are independent of the choice of s. Then, what we established in the previous subsection, the estimate obtained just above and  

  (•) (•) lib (N )(·), U N ((·) ∨ s)U N (s)∗ Fs EW [X [Pk ] N ] = E Pk ξ•  

(s)  = E Pk,N Fs , 1 ≤ k ≤ m   s  s  altogether force τ (E(P1 ; τ ) · · · E(Pm ; τ )) to be τ E sτ P1τ · · · E sτ Pmτ , and we finally obtain that      

       tr N E P (s) Fs · · · E P (s) Fs − τ E τ P τ s · · · E τ P τ s  s s m 1 1,N m,N   ≤

  C1 + C2 d τlib (N ) , τ . 2 N

Since the right-hand side is independent of the choice of s, the desired uniform (in time s) convergence follows.   4.3 A Slight Generalization of [13, Proposition 3.5] ±1 Let w = w(1) · · · w(r ) be a word in the letters d1 , . . . , dr and u ±1 1 , . . . , u r . Define

εk :=

+1 (w(k) = d∗ or w(k) = u ∗ = u +1 ∗ ), −1 (w(k) = u −1 ∗ )

for 1 ≤ k ≤ r . In what follows, we may regard k → w(k) as a function from {1, . . . , r } to the letters di , u i±1 . Let U N(i) , i = 1, 2, . . . , be independent left unitary Brownian motions as before, and Di ∈ M N (C), i = 1, 2, . . . , be given matrices. The substitution of Di and U N(i) (ti ) for di and u i , respectively, into w is denoted by   (•) w D• , U N (t• ) = W N = W N (1) · · · W N (r ) (i)

(whose values are taken in M N (C)) with W N (k) = Di or W N (k) = U N (ti )±1 . Moreover, we set   w⊗ D• , U N(•) (t• ) = W N⊗ = W N (1) ⊗ · · · ⊗ W N (r ) (whose values are taken in M N (C)⊗r ).

123

J Theor Probab

With the permutation representation ρ : C[Sr ]  (C N )⊗r (see Sect. 4.2.2), we write   1 ⊗ ⊗r p N (t; σ ) := E Tr (ρ(σ )W N ) N (σ ) with t = (t1 , . . . , tr ). (n.b. (σ ) denotes the number of cycles in σ as before.) The family p N (t; σ ), σ ∈ Sr , forms an r ! dimension column vector p N (t) with indices ε,δ on Sr , 1 ≤ l, m ≤ r , ε, δ ∈ {±1}, defined by Sr . We introduce the operation l,m

ε,δ l,m (σ ) :=

⎧ ⎪ ⎪σ (l, m) ⎪ ⎪ ⎪ ⎨(l, m)σ

(ε = δ = +1), (ε = δ = −1),

⎪ ⎪ (σ (l), m)σ (ε = +1, δ = −1), ⎪ ⎪ ⎪ ⎩ (σ (m), l)σ = σ (σ −1 (l), m) (ε = −1, δ = +1).

A tedious calculation confirms that 







ε,δ ,δ ε ,δ ε,δ   l,m ◦ lε ,m  = l  ,m  ◦ l,m as long as {l, m} ∩ {l , m } = ∅

(19)

for any choice of ε, ε , δ, δ  . We also define the r ! × r ! matrices Ai (w) (with indices Sr ) by setting the (σ, σ  )-entry as  1 Ai (w)σ,σ  := − w −1 ({u i± }) δσ,σ  − 2

 l,m∈w−1 ({u i± }) σ

εl εm δεl ,εm (σ ),σ  , l,m

l
σ

where |w −1 ({u i± })| denotes the number of elements of w −1 ({u i± }) and l ∼ m means that both l, m are in a common cycle of σ . Then, the matrices Ai (w) mutually commute, since the w −1 ({u i±1 }) are disjoint. In what follows,  − ∞ means the ∞ -norm on the r !-dimensional vector space of column vectors. The next proposition is just a slight generalization of [13, Proposition 3.5], whose proof is a reorganization of the original one. Proposition 4.6 With γr := (1, 2, . . . , r ) ∈ Sr , we have  

  E tr N (w(D• , U (•) (t• ))) N    -, , r     1 ⊗r ti Ai (w) Tr N (ρ(σ )w⊗ (Di , I N )) exp − (σ ) N  i=1 σ ∈Sr γ −1 ,σ   , r -r      ≤  p N (t) − exp ti Ai (w) p N (0)   i=1

123

∞

J Theor Probab

1 ≤ 2N 2

, r  6 72 %r 6 7     −1 u ±1  2 ±1 −1 i=1 ti w i    p N (0)∞ ui ti w e i=1

r 3T r3T  p N (0)∞ ≤ e 2N 2 with 0 = (0, . . . , 0) and T := max1≤i≤r ti , and furthermore , , r --   6 72   1 %r  −1 u ±1    r3T i=1 ti w i 2  exp  ≤ e t A (w) ≤e 2 . i i     i=1 σ,σ  ⊗r 1 Remark that N (σ ) Tr N (ρ(σ )w⊗ (D• , I N )) is a product of moments in the Di with respect to tr N of degree less than r . Hence, the above proposition (together with the method in the previous subsection) strengthens Biane’s asymptotic freeness result [2, Theorem 1(2)] for left unitary Brownian motions with constant matrices in the fashion that the convergence as N → ∞ is uniform on finite time intervals.

Proof (A reproduction of the proof of [13, Proposition 3.5]) The algebra M N (C)⊗r has r different M N (C)-bimodule structures (+1)

θk

(−1)

θk

M N (C)  M N (C)⊗r  M N (C) defined by (+1)

θk

(X )(Y1 (−1) θk (X )(Y1

⊗ · · · ⊗ Yk ⊗ · · · ⊗ Yr ) := Y1 ⊗ · · · ⊗ X Yk ⊗ · · · ⊗ Yr , ⊗ · · · ⊗ Yk ⊗ · · · ⊗ Yr ) := Y1 ⊗ · · · ⊗ Yk X ⊗ · · · ⊗ Yr

for X ∈ M N (C) and Y1 ⊗ · · · ⊗ Yr ∈ M N (C)⊗r . The Itô formula enables us to obtain (see [13, Lemma 3.7]) that ∂ p N (t; σ ) ∂ti

 1 = − w −1 ({u i± }) p N (t; σ ) + 2

 6 7 u i±1 l
l,m∈w−1

εl εm

1 N (σ )

E

 

× Tr N (θl−εl ⊗ θm−εm )(Cu(N ) )ρ(σ )W N⊗ , (20) %N where Cu(N ) = − N1 α,β=1 E αβ ⊗ E βα with matrix units E αβ for M N (C). Then, by [13, Lemmas 3.8 and 3.9] we have

123

J Theor Probab

 

−εl ⊗ −εm (θ E Tr ⊗ θ )(C )ρ(σ )W N u(N ) m l N ) N (σ⎧ σ ε ,ε ⎨− p N (t;  l m (σ )) (l ∼ m), l,m = σ ⎩− 1 p (t; εl ,εm (σ )) (l  m). l,m N2 N 1

Therefore, with the r ! × r ! matrices Ci (w) (with indices Sr ): Ci (w)σ,σ  := −

 l,m∈w−1 ({u i± })

εl εm δεl ,εm (σ ),σ  , l,m

σ

l
we can rewrite (20) as ∂ p N (t) = ∂ti

.

/ 1 Ai (w) + 2 Ci (w) p N (t) (i = 1, . . . , r ), N

which implies that , p N (t) = exp

 i

1  ti Ai (w) + 2 ti Ci (w) p N (0), N i

since the Ai (w) and the Ci (w) mutually commute due to (19). Let  −  denote the operator norm with respect to  − ∞ on the r !-dimensional vector space of column vectors. Observe that

Ci (w) ≤

  −1 6 ±1 7   −1 6 ±1 7   w −1 w ui ui

≤

1  −1 6 ±1 7 2 w ui , 2

2 1  −1 6 ±1 7 2 Ai (w) ≤ w ui . 2 % % Write A := i ti Ai (w) and C := i ti Ci (w) for simplicity. Then, we have

 p N (t) − (exp A) p N (0)∞   . /   1  ≤ exp A + 2 C − exp A   p N (0)∞ N  1 . . . // /    1 d exp s A + = C exp((1 − s)A) ds     p N (0)∞ 2 N 0 ds  . . / . 1  //    1  1     ≤ exp s A + N 2 C   N 2 C  exp((1 − s)A) ds  p N (0)∞ 0 . /  1 s 1 C A N2 ≤ C e e ds  p N (0)∞ N2 0

123

J Theor Probab

1 C eA+C  p N (0)∞ N2 , , r r 6 7      −1 6 ±1 7 2 1 2 ±1 −1     ui ui ≤ ti w ti w exp  p N (0)∞ . 2N 2 ≤

i=1

i=1

 

Hence, we are done.

5 Large Deviation Upper Bound This section is concerned with the proof of the desired large deviation upper bound for τlib (N ) . To this end, we prove in Sect. 5.1 the exponential tightness of the sequence of probability measures P(τlib (N ) ∈ · ), and then, in Sect. 5.2, introduce and investigate an appropriate rate function by looking at Proposition 3.2. In Sect. 5.3, with these preparations, we finalize the proof by using Theorem 4.1 (with Proposition 2.3). 5.1 Exponential Tightness Let us start with the next exponential estimate for left unitary Brownian motions. This lemma is inspired by the proof of [4, Lemma 2.5]. Proposition 5.1 Let U N be an N × N left unitary Brownian motion as in the introduction. Then, -

, P

sup s≤t≤s+δ

U N (s) − U N (t)tr N ,2 ≥ ε

√ 2 2 ≤ 2 2 e−N L(ε −(8L+1)δ)

holds for every s ≥ 0, ε > 0, δ > 0 and L > 0. Proof With Z N (t) := tr N (2Re(I N − U N (t))), we observe that -

, P

sup s≤t≤s+δ

U N (s) − U N (t)tr N ,2 ≥ ε

,

=P

sup U N (s) − U N (s ,

0≤t≤δ

sup U N (s + t)U N (s) − ,

0≤t≤δ

sup U N (t) −

=P ,

0≤t≤δ

sup Z N (t) ≥ ε

≥ε

2

∗

=P

=P

+ t)2tr N ,2

I N 2tr N ,2

≥ε

2

I N 2tr N ,2

≥ε

2

2

0≤t≤δ

123

J Theor Probab

by the left increment property of left unitary Brownian motions. Thus, it suffices to estimate P(sup0≤t≤δ Z N (t) ≥ ε2 ) from the above. One has  t 2Re(I N − U N (t)) = − i (dH N (s)U N (s) − U N (s)∗ dH N (s)) 0  t + Re(U N (s)) ds, 0

since dU N (t) = i dH N (t) U N (t) − 21 U N (t) dt with N × N self-adjoint Brownian motion H N . Set 8N (t) := − M



t

i (dH (s)U N (s) − U N (s)∗ dH (s))  t = 2 Re(I N − U N (t)) − Re(U N (s)) ds, 0

0

9 8N (t)) = Z N (t) − t Re(tr N (U N (s))) ds defines a and observe that M N (t) := tr N ( M 0 martingale. Let Cαβ be the standard orthogonal basis of M N (C)sa as in the introduc%N Bαβ (t) √ tion. Then, H N (t) = α,β=1 Cαβ with an N 2 -dimensional standard Brownian N motion Bαβ . This expression enables us to compute the quadratic variation N  t  2 1  M N (t) = 3 Tr N i (Cαβ U N (s) − U N (s)∗ Cαβ ) dt N 0 α,β=1

N  t  2 1  Tr N i (U N (s) − U N (s)∗ )Cαβ dt 3 N α,β=1 0  t   1 4t i(U N (s) − U N (s)∗ )2 = 2 dt ≤ 2 tr N ,2 N 0 N

=

as in Sect. 3. 9t Note that Z N (t) = M N (t) + 0 Re(tr N (U N (s))) ds ≤ |M N (t)| + t. Hence, if sup0≤t≤δ Z N (t) ≥ ε2 , then we have both sup0≤t≤δ |M N (t)| ≥ ε2 − δ and sup exp(−N 2 L M N (t)) + sup exp(N 2 L M N (t)) 0≤t≤δ

0≤t≤δ

  ≥ sup exp(−N 2 L M N (t)) + exp(N 2 L M N (t)) 0≤t≤δ

≥ sup exp(N 2 L|M N (t)|) ≥ e N 0≤t≤δ

123

2 L(ε 2 −δ)

J Theor Probab

for any fixed L > 0. Thus, we get , , sup Z N (t) ≥ ε

P

≤P

2

0≤t≤δ

≤e

sup |M N (t)| ≥ ε − δ 2

0≤t≤δ

:

−N 2 L(ε2 −δ)

E

sup exp(−N L M N (t)) 0≤t≤δ

: +E

; 2

;<

sup exp(N L M N (t)) 2

0≤t≤δ

by Chebyshev’s inequality. We have :

;

E sup exp(±N 2 L M N (t)) 0≤t≤δ :

. . / . //; 1 1 2 2 2 = E sup exp ±N L M N (t) − ±N L M N (t) exp ±N L M N (t) 2 2 0≤t≤δ : . / . /; 1 1 2 2 2 ±N L M N (δ) ≤ E sup exp ±N L M N (t) − ±N L M N (t) × exp 2 2 0≤t≤δ : . /2 ;1/2 1 2 2 ≤ E sup exp ±N L M N (t) − ±N L M N (t) 2 0≤t≤δ : ; . /2 1/2 1 ±N 2 L M N (δ) × E exp 2 : . /2 ;1/2 1 2 2 ≤ E sup exp ±N L M N (t) − ±N L M N (t) 2 0≤t≤δ

 1/2 × E exp N 4 L 2 M N (δ)

 by the Cauchy–Schwarz inequality. Since t → exp ±N 2 L M N (t) − 21 ±N 2 L M N    (t)) and t → exp ±4N 2 L M N (t) − 21 ±4N 2 L M N (t) are martingales thanks to [11, Corollary 3.5.13], Doob’s maximal inequality with ‘ p = 2’ (see e.g., [11, Theorem 1.3.8(iv)] with the help of Jensen’s inequality) shows that : . /2 ; 1 2 2 E sup exp ±N L M N (t) − ±N L M N (t) 2 0≤t≤δ : /2 ; . 1 ≤ 2 E exp ±N 2 L M N (δ) − ±N 2 L M N (δ) 2

  = 2 E exp ±2N 2 L M N (δ) − 4±N 2 L M N (δ) + 3±N 2 L M N (δ)   2 1/2 ≤ 2 E exp ±2N 2 L M N (δ) − 4±N 2 L M N (δ)

123

J Theor Probab

  2 1/2 2 × E exp 3±N L M N (δ)  . /1/2 1 = 2 E exp ±4N 2 L M N (δ) − ±4N 2 L M N (δ) 2

 1/2 × E exp 6±N 2 L M N (δ)

 1/2 = 2 E exp 6N 4 L 2 M N (δ) . Therefore, we have : E

;

sup exp(±N L M N (t)) 2

0≤t≤δ

≤

 1/4

 1/2 √ √ 2 2 2 E exp 6N 4 L 2 M N (δ) E exp N 4 L 2 M N (δ) ≤ 2 e8N L δ .

Hence, we get , P

≤ e−N

sup Z N (t) ≥ ε2

2 L(ε 2 −δ)

×2×

√

2 e8N

2 L2δ

√ 2 2 = 2 2 e−N L(ε −(8L+1)δ)

0≤t≤δ

for every L > 0.

 

Corollary 5.2 The sequence of probability measures P(τlib (N ) ∈ · ) on T S c  ∗  C R x• (·) is exponentially tight. Proof Observe that  1/2 max τlib (N ) (xi j (s) − xi j (t))2

sup

0≤s,t≤k 1≤ j≤r (i) |s−t|≤δ 1≤i≤n+1

≤

≤

max

sup

 1/2 τlib (N ) (xi j (s) − xi j (t))2

max

sup

.  1/2 τlib (N ) (xi j (s) − xi j ( δ))2

0≤ ≤[k/δ] δ≤s≤( +1)δ 1≤ j≤r (i) s≤t≤s+δ 1≤i≤n

0≤ ≤[k/δ] δ≤s≤( +1)δ 1≤ j≤r (i) s≤t≤s+δ 1≤i≤n



+ τlib (N ) (xi j ( δ) − xi j (t)) ≤2

sup

0≤ ≤[k/δ] δ≤t≤( +2)δ 1≤ j≤r (i) 1≤i≤n

≤ 4R

123

max

max

sup

2

1/2 /

 1/2 τlib (N ) (xi j ( δ) − xi j (t))2

0≤ ≤[k/δ] δ≤t≤( +2)δ 1≤i≤n

   (i)  (i) U N ( δ) − U N (t)

tr N ,2

,

J Theor Probab

where is a parameter of nonnegative integers and [k/δ] denotes the greatest nonnegative integer that is not greater than k/δ. Hence, for each k ∈ N and for any δ > 0 and L > 0, we have ⎞ ⎛  1/2 ⎟ ⎜ max τlib (N ) (xi j (s) − xi j (t))2 > 1/k ⎠ P ⎝ sup 0≤s,t≤k 1≤ j≤r (i) |s−t|≤δ 1≤i≤n+1

⎛

≤ P ⎝ max

sup

0≤ ≤[k/δ] δ≤t≤( +2)δ 1≤i≤n

≤

[k/δ] n 

,

P

sup

δ≤t≤( +2)δ

=0 i=1

    (i) (i) U N ( δ) − U N (t)

tr N ,2

    (i) (i) U N ( δ) − U N (t)

tr N ,2

⎞ 1 ⎠ ≥ 4Rk

1 ≥ 4Rk

-

√ 2 2 2 −1 ≤ 2 2 n ([k/δ] + 1) e−N L((16R k ) −(8L+1)2δ) by Proposition 5.1. Therefore, for a given C > 0, letting L := 32R 2 k 3 C and δk :=

1 , 64R 2 k 2 (256R 2 k 3 C + 1)

we obtain the following estimate: ⎛

⎞  1/2 1⎟ ⎜ P ⎝ sup max τlib (N ) (xi j (s) − xi j (t))2 > ⎠ k 0≤s,t≤k 1≤ j≤r (i) |s−t|≤δk 1≤i≤n+1

≤ C  (k 6 e−N

2C

k/2

) e−N

2C

k/2

,

where C  > 0 depends only on n, R, C and is independent of k, N . If C > 12, then 2 2 k 6 e−N Ck/2 ≤ e−N C/2 . With the sequence (δk )k≥1 , it follows that   / (δk ) P τlib (N ) ∈ ⎛ ∞  ⎜ ≤ P ⎝ sup k=1

≤C



⎞  1/2 1⎟ max τlib (N ) (xi j (s) − xi j (t))2 > ⎠ 1≤ j≤r (i) k

0≤s,t≤k |s−t|≤δk 1≤i≤n+1

e−N

2C

1 − e−N

2 C/2

,

  implying that lim N →∞ N12 log P τlib (N ) ∈ / (δk ) ≤ −C whenever C > 12. This together with Lemma 2.2(2) shows the exponential tightness of the measures   P(τlib (N ) ∈ · ), since C > 0 can arbitrarily be large.

123

J Theor Probab

5.2 Rate Function   We define a map Iσlib : T S c C ∗R x• (·) → [0, +∞] to be 0 τ

sup

T ≥0

P=P ∗ ∈Cx

1 2 n

T

(P) − σ0lib (P) −

• (·)



T

k=1 0

   s 2  τ  τ x• (·), v•τ (·)  E s D(k) s P

τ,2

< ds .

(21) That the integrand is piecewise continuous in s follows from Lemma 5.5 below (k) together with (22): Note that i Ds P is self-adjoint if P = P ∗ , and then    s 2  τ  τ τ P x (·), v (·) E s D(k)  s • • τ,2    s 2  τ  (k) τ τ = E s (i Ds P) x• (·), v• (·)  τ,2    2  τ = −τ E sτ D(k) v•τ ((· − s) ∨ 0)x• (· ∧ s)v•τ ((· − s) ∨ 0)∗ , v•τ (·) s P (22) holds for every P = P ∗ ∈ C x• (·). Lemma 5.3 If Iσlib (τ ) < +∞, then τ 0 = σ0lib , that is, π0∗ (τ ) = σ0 , and 0

Iσlib (τ ) = 0

1 2

⎧ ⎪ ⎨

sup

T ≥0

P=P ∗ ∈Cx

•

 T 2 τ (P) − σ0lib (P)    2 ⎪ (k) τ s (·), v τ (·)  ⎩ %n 9 T  τ D P x E  s • s • k=1 0 (·)

τ,2

⎫ ⎪ ⎬ ⎪ ds ⎭

holds (and the right-hand side is well defined with convention 0/0 = 0, that is, if the denominator is zero, then the numerator must be zero). Proof For each fixed P = P ∗ ∈ C x• (·), let αT (P) := τ T (P) − σ0lib (P) and   s 9T  2 %  (k) τ (·), v τ (·)  βT (P) := nk=1 0  E sτ Ds P x•  ds, and consider the func• τ,2

tion f P,T (r ) := αT (r P) −

βT (r P) βT (P) 2 = αT (P) r − r 2 2 . / αT (P) 2 αT (P)2 βT (P) r− =− + 2 βT (P) 2βT (P)

on the real line. If βT (P)  0, then maxr f P,T (r ) = f P,T (αT (P)/βT (P)) = αT (P)2 /2βT (P); otherwise, sup f P,T (r ) = sup αT (P)r = r

123

r

0

(αT (P) = 0),

+∞ (αT (P) = 0).

J Theor Probab

Trivially, β0 (P) = 0 always holds, and hence the above discussion shows that (τ ) < +∞. Therefore, we have proved the α0 (P) must be 0 for every P, since Iσlib 0 former assertion τ 0 = σ0lib . (τ ) − ε < For any ε > 0, there exist Pε = Pε∗ ∈ C x• (·) and Tε ≥ 0 so that Iσlib 0 lib f Pε ,Tε (1) ≤ maxr f Pε ,Tε (r ) ≤ Iσ0 (τ ) < +∞. Then, the first paragraph shows that Iσlib (τ ) − ε < 0

αT (P)2 αTε (Pε )2 ≤ sup = sup max f P,T (r ) ≤ Iσlib (τ ) 0 2βTε (Pε ) P,T 2βT (P) P,T r

with convention 0/0 = 0. Hence, the latter assertion holds.

 

Here is a simple lemma. Lemma 5.4 Let (M, τ ) be a tracial W ∗ -probability space with τ faithful, and u ∈ M be a unitary, and N be a (unital) W ∗ -subalgebra of M. Let E : M → N be the unique τ -preserving conditional expectation. If u is ∗-freely independent of N , we have E(uxu ∗ ) = τ (x)1 + |τ (u)|2 x ◦ for every x ∈ N with x ◦ := x − τ (x)1. Proof For every y ∈ N , we have τ (uxu ∗ y) = τ (x)τ (y) + |τ (u)|2 τ (x ◦ y) by the ∗-free independence between u and N . Since E(uxu ∗ ) ∈ N is uniquely determined by the relation τ (uxu ∗ y) = τ (E(uxu ∗ )y) for every y ∈ N , the desired assertion immediately follows.   The same idea as above shows the next lemma. Lemma 5.5 Let (M, τ ) be a tracial W ∗ -probability space with τ faithful. Let L and N be freely independent (unital) W ∗ -subalgebras of M, and E : M → N be the unique τ -preserving conditional expectation. Then, ((a1 , . . . , an−1 , an ), (b1 , . . . , bn−1 )) ∈ Ln × N n−1 → E(a1 b1 · · · an−1 bn−1 an ) ∈ N is written as a universal polynomial in moments of the ai , moments of the bi and words in the bi . Proof Let us calculate the map ((a1 , . . . , an−1 , an ), (b1 , . . . , bn−1 , bn )) ∈ Ln × N n → τ (a1 b1 · · · an−1 bn−1 an bn ). By [15, Proposition 11.4, Theorem 11.16], τ (a1 b1 · · · an−1 bn−1 an bn ) is a universal polynomial in moments of the ai and moments of the bi . Since the map ((a1 , . . . , an−1 , an ), (b1 , . . . , bn−1 , bn )) → τ (a1 b1 · · · an−1 bn−1 an bn ) is multilinear, each term of the polynomial includes some joint moments of the bi , where bn appears only once in a unique joint moment. Then, we can obtain the desired assertion in the same way as in the proof of Lemma 5.4.   We remark that the universal polynomial whose existence we have established admits an explicit formula based on the notation in [15, Lecture 11]. Here is a main result of this subsection.

123

J Theor Probab

  Proposition 5.6 Iσlib : T S c C ∗R x• (·) → [0, +∞] is a good rate function. 0 Proof By (22) together with Lemma 5.5, we observe that    s 2   τ τ τ → E sτ D(k) P x (·), v (·)  s • •

τ,2

is a continuous function for every s. Hence, τ → I P,T (τ ) 1 2 n

:= τ T (P) − σ0lib (P) −



k=1 0

T

   s 2  τ  τ τ P x (·), v (·) E s D(k)  s • •

τ,2

ds

is continuous, and consequently, Iσlib is lower semicontinuous. Therefore, it suffices to 0 lib prove that the level set {Iσ0 ≤ λ} sits in a compact subset for every nonnegative real number λ ≥ 0. (τ ) ≤ λ. By Lemma 5.3 we have Assume that Iσlib 0 = > n  >  T lib τ (P) ≤ σ (P) + ?2λ 0

k=1 0

T

   2  τ (k) τ s (·), v τ (·)  E s Ds P x•  •

τ,2

ds

(23)

for every P = P ∗ ∈ C x• (·) and T ≥ 0. For 0 ≤ t1 < t2 we have   0 2 (x = 2δk,i 1[0,t1 ] (s)vi (t1 − s)∗ [xi j (t1 ), xi j (t2 )]vi (t1 − s) (t ) − x (t )) D(k) i j 1 i j 2 s 1 +1[0,t2 ] (s)vi (t2 − s)∗ [xi j (t2 ), xi j (t1 )]vi (t2 − s) , and hence   2 τs τ D(k) s ((x i j (t1 ) − x i j (t2 )) ) (x i j (·), vi (·)) 6 ⎧ ⎪ 2δk,i [xiτj (s), viτ (t1 − s)∗ viτ (t2 − s)xiτj (s)viτ (t2 − s)∗ viτ (t1 − s)] ⎪ ⎪ 7 (s ≤ t1 ), ⎪ ⎪ ⎪ ⎨ + [xiτj (s), viτ (t2 − s)∗ viτ (t1 − s)xiτj (s)viτ (t1 − s)∗ viτ (t2 − s)] = ⎪ ⎪ (t1 < s ≤ t2 ), 2δk,i [xiτj (s), viτ (t2 − s)∗ xiτj (t1 )viτ (t2 − s)] ⎪ ⎪ ⎪ ⎪ ⎩ 0 (t2 < s).

When s ≤ t1 , Lemma 5.4 enables us to compute    2 τs τ D(k) s ((x i j (t1 ) − x i j (t2 )) ) (x i j (·), vi (·)) 6

  = 2δk,i xiτj (s), E sτ viτ (t1 − s)∗ viτ (t2 − s)xiτj (s)viτ (t2 − s)∗ viτ (t1 − s)

E sτ

123

J Theor Probab



7 + xiτj (s), E sτ viτ (t2 − s)∗ viτ (t1 − s)xiτj (s)viτ (t1 − s)∗ viτ (t2 − s) 6

   2 = 2δk,i xiτj (s), τ (xiτj (s))1 + τ (viτ (t1 − s)∗ viτ (t2 − s)) xiτj (s)◦

7   2 + xiτj (s), τ (xiτj (s))1 + τ (viτ (t2 − s)∗ viτ (t1 − s)) xiτj (s)◦ = 0. In this way, we obtain the formula: E sτ

   2 τs τ D(k) s ((x i j (t1 ) − x i j (t2 )) ) (x i j (·), vi (·))

(24)

= 2δk,i 1(t1 ,t2 ] (s)|τ (viτ (t2 − s))|2 [xiτj (s), xiτj (t1 )◦ ].

Then, (23) with P := (xi j (t1 ) − xi j (t2 ))2 and T large enough, and (24) altogether show that @ τ ((xi j (t1 ) − xi j (t2 ))2 ) ≤ σ0lib ((xi j (t1 ) − xi j (t2 ))2 ) + 8R 2 2λ|t1 − t2 |. By the construction of σ0lib (see Sect. 2), we see that σ0lib ((xn+1 j (t1 )−xn+1 j (t2 ))2 ) = 0 and moreover that, if 1 ≤ i ≤ n, then σ0lib



xi j (t1 ) − xi j (t2 )

2 

 2   = vi (t1 )xiσj0 vi (t1 )∗ − vi (t2 )xiσj0 vi (t2 )∗  σ˜ 0 ,2 2  ≤ 2Rvi (t1 ) − vi (t2 )σ˜ 0 ,2 = 4R 2 vi (|t1 − t2 |) − 12σ˜ 0 ,2 → 0

as |t1 − t2 | → 0. Hence, by Lemma 2.2(2), {Iσlib ≤ λ} sits inside a compact subset.   0 We give a few important properties on the rate function Iσlib . 0   Proposition 5.7 For any τ ∈ T S c C ∗R x• (·) , we have: τ (1) Iσlib (τ ) < +∞ implies that t → xn+1 j (t) is a constant process for every 1 ≤ 0 j ≤ r (n + 1). (τ ) < +∞ implies that for each fixed 1 ≤ i ≤ n and t ≥ 0, we have (2) Iσlib 0 πt∗ (τ )(P) = σ0 (P) for every non-commutative polynomial P in indeterminates xi j , 1 ≤ j ≤ r (i). (τ ) = 0 if and only if τ = σ0lib . Hence, σ0lib is a unique minimizer of Iσlib . (3) Iσlib 0 0 σ lib

τ τ 0 2 Proof (1) By (23) and (24), we have xn+1 j (t) − x n+1 j (0)τ,2 ≤ x n+1 j (t) − σ lib

σ0 σ0 τ τ 0 2 2 xn+1 j (0)σ lib ,2 = x n+1 j − x n+1 j σ0 ,2 = 0. Hence, x n+1 j (t) = x n+1 j (0) holds for 0

every t ≥ 0. (2) Let P be an arbitrary, non-commutative polynomial in indeterminates xi j , 1 ≤ (k) j ≤ r (i), with a fixed 1 ≤ i ≤ n. It is easy to see that Ds πt (P) = 0. Hence, we have

123

J Theor Probab

  r πt∗ (τ T )(P) − πt∗ (σ0lib )(P) = τ T (r πt (P)) − σ0lib (r πt (P)) ≤ Iσlib (τ ) < +∞ 0 for every r ∈ R and T ≥ 0, and thus πt∗ (τ )(P) = πt∗ (τ T )(P) = πt∗ (σ0lib )(P) = σ0 (P) with T large enough. (3) By the left increment property of left free unitary Brownian motions (see [2, Definition 2]), it is easy to see that (σ0lib )T = σ0lib holds for every T ≥ 0. Thus, we trivially obtain that (σ0lib ) Iσlib 0

=

sup

T ≥0 P=P ∗ ∈Cx• (·)

 −

T 0

n     σ0lib  (k)   τ s 2 τ  Es  D P x (·), v (·) s • •   lib

< ds

σ0 ,2

k=1

= 0. Lemma 5.3 with its proof actually shows that Iσlib (τ ) = 0 implies that 0 2  T τ (P) − σ0lib (P) 0≤   9T  2 %  τ s (·), v τ (·)  2 nk=1 0  E sτ D(k) x•  s P •

τ,2

(τ ) = 0 ≤ Iσlib 0 ds

(with convention 0/0 = 0) for all P = P ∗ ∈ C x• (·) and T ≥ 0. This (with the proviso in Lemma 5.3) actually shows that τ T (P) = σ0lib (P) holds for every   P = P ∗ ∈ C x• (·) and T ≥ 0. This immediately implies that τ = σ0lib . These properties actually show that Iσlib is indeed a ‘right’ rate function for our 0 will be given in a sequel to this purpose. Further analysis of this rate function Iσlib 0 article. 5.3 Main Results We are ready to prove the next main result of this article.   Theorem 5.8 For every closed subset of T S c C ∗R x• (·) , we have lim

N →∞

7 6    1 log P τlib (N ) ∈ ≤ − inf Iσlib (τ )  τ ∈ . 0 2 N

Proof Since the P(τlib (N ) ∈ ·) form an exponentially tight sequence of probability measures and Iσlib is a good rate function, it suffices to prove the following weak large 0 deviation upper bound: lim lim

ε0 N →∞

    1 log P d τlib (N ) , τ < ε ≤ −Iσlib (τ ) 0 2 N

  for every τ ∈ T S c C ∗R x• (·) . (This is a standard fact in large deviation theory; see the proofs of [6, Theorem 4.1.11, Lemma 1.2.18].)

123

J Theor Probab

Consider the random variable     I P,T,N := E τlib (N ) (P) | FT − E τlib (N ) (P) n 

   2  1 T   (•) (•) lib ∗  − P ξ (N ), U (· + s)U (s) ds. Fs  E D(k) s • N N tr N ,2 2 0 k=1

By Proposition 3.2 we have E[exp(N 2 I P,T,N )] = E[exp(N 2 I P,0,N )] = 1.

(25)

Let I P,T (τ ) be as in the proof of Proposition 5.6. We have       2  7 exp(N 2 I P d τlib (N ) , τ < ε = E 16  − N I ) P,T,N P,T,N d τlib (N ) ,τ <ε    7 exp(N 2 I ≤ E 16  P,T,N ) d τlib (N ) ,τ <ε 7 6    × esssup exp(−N 2 I P,T,N )  d τlib (N ) , τ < ε 6    7 ≤ esssup exp(−N 2 I P,T,N )  d τlib (N ) , τ < ε (use (25))     0  1 = exp −N 2 essinf I P,T,N  d τlib (N ) , τ < ε . Observe that I P,T,N ≥ I P,T (τ ) − |I P,T,N − I P,T (τ )|   0  1 ≥ I P,T (τ ) − esssup |I P,T,N − I P,T (τ )|  d τlib (N ) , τ < ε  1 0  holds almost surely on d τlib (N ) , τ < ε . Therefore, we conclude that     1 log P d τlib (N ) , τ < ε ≤ −I P,T (τ ) 2 N    1 0 + esssup |I P,T,N − I P,T (τ )|  d τlib (N ) , τ < ε . Then, Proposition 2.3 and Corollary 4.2 (together with (3) and (22)) show that 6 7     lim lim esssup  I P,T,N − I P,T (τ )  d τlib (N ) , τ < ε = 0,

ε0 N →∞

and hence lim lim

ε0 N →∞

    1 log P d τlib (N ) , τ < ε ≤ −I P,T (τ ) 2 N

for every P = P ∗ ∈ C x• (·) and T ≥ 0. Hence, we are done.

 

123

J Theor Probab

Here is a standard application of the above large deviation upper bound and Proposition 5.7(3).   Corollary 5.9 We have lim N →∞ d τlib (N ) , σ0lib = 0 almost surely. Proof Let ε > 0 be arbitrarily chosen. By Propositions 5.6 and 5.7(3), we observe that inf{Iσlib (τ ) | d(τ, σ0lib ) ≥ ε}  0. Then, Theorem 5.8 implies that 0 lim

N →∞

   6   7 1  log P d τlib (N ) , σ0lib ≥ ε ≤ − inf Iσlib (τ )d(τ, σ0lib ) ≥ ε  0. 0 2 N

% lib Thus, we obtain that ∞ N =1 P(d(τlib (N ) , σ0 ) ≥ ε) < +∞. Hence, the desired assertion follows by the Borel–Cantelli lemma.  

6 Discussions One of the motivations in mind is to provide a common basis for the study of Voiculescu’s approach [22] and our orbital approach [8,19] to the concept of mutual information in free probability. In fact, the key ingredient of Voiculescu’s approach is the liberation process, while the orbital approach involves ‘orbital microstates’ by unitary matrices. Thus, a serious lack was a random matrix counterpart of liberation process, whose candidate we introduced in this article. Here, we are not going to any detailed discussions about such a study, but only give some comments on it. We may apply the contraction principle in large deviation theory to our large deviation upper bound obtained in Sect. 5. Corollary 6.1 Let ν N ,T be the marginal probability distribution on U(N ) of the N × N left unitary Brownian motion at time T > 0. Define  6 7  ∗ lib π (σ ) := inf I (τ ) (τ ) = σ , Iσlib  σ0 T 0 ,T

  σ ∈ T S C ∗R x•  .

  Then, for any closed subset of T S C ∗R x•  we have lim

N →∞

  7 6 7 6 1  ⊗n n  (N ) lib tr U ∈ U(N ) log ν ∈

≤ − inf I (σ ) σ ∈ .  σ0 ,T N ,T U 2 N

  (N ) (N ) n Here, tr U ∈ T S C ∗R x•  with U = (Ui )i=1 ∈ U(N )n is defined by tr U (P) := tr N (U (P)), P ∈ C x• , where U : C x•  → M N (C) is a unique ∗homomorphism sending xi j (1 ≤ i ≤ n) to Ui ξi j (N )Ui∗ and xn+1 j to ξn+1 j (N ). We write T χorb (σ ) := lim

lim

m→∞ N →∞ δ0

123

 6 7 1 ⊗n n  (N ) tr U ∈ U(N ) log ν ∈ O (σ ) ,  m,δ N ,T U N2

J Theor Probab

  where Om,δ ), m ∈ N, δ > 0, denotes the (open) subset of σ  ∈ T S C ∗R x•   (σ  such that σ  (xi1 j1 · · · xi p j p ) − σ (xi1 j1 · · · xi p j p ) < δ whenever 1 ≤ i k ≤ n + 1, 1 ≤ jk ≤ r (i k ), 1 ≤ k ≤ p and 1 ≤ p ≤ m. T (σ ) A problem in this direction is to show that χorb (σ ) ≤ lim T →+∞ χorb holds, where χorb (σ ) denotes the orbital free entropy of the random multi-variables (xi j )1≤ j≤r (i) , 1 ≤ i ≤ n, under σ (see [8,19]). If this was the case, then T (σ ) (see below) and χ (σ ) ≤ we would obtain that χorb (σ ) = lim T →+∞ χorb orb lib − lim T →+∞ Iσ0 ,T (σ ). Remark that, if the families {xi j }1≤ j≤r (i) , 1 ≤ i ≤ n, are freely independent under σ0 , then it is easy to see that πT∗ (σ0lib ) = σ0 for all T ≥ 0, (σ0 ) = 0 for all T ≥ 0 so that and hence Proposition 5.7(3) shows that Iσlib 0 ,T (σ ) holds as 0 = 0 for all T ≥ 0. Thus, our conjecture seems χorb (σ0 ) = −Iσlib 0 0 ,T plausible. Here, we would like to point out that 3 2  dν N ,T 1  lim lim log max (U )  U ∈ U(N ) T →+∞ N →∞ N 2 dν N 1 dν N ,T = lim lim log (I N ) = 0 T →+∞ N →∞ N 2 dν N with the Haar probability measure ν N on U(N ) follows from the formula obtained precisely by Lévy and Mäida [14, Proposition 4.2; Lemma 4.7; Proposition 5.2] with the aid of the fact that 

1

K (k) = 0

@

ds (1 − s 2 )(1 − k 2 s 2 )

3 1 = − log(1 − k) + log 2 + o(1) (as k  1). 2 2

  Thus, for any Borel subset of T S C ∗R x•  we have 6 7  1 ⊗n n  (N ) tr U ∈ U(N ) log ν ∈

N ,T U N2 6 7  (N ) n dν N ,T 1 ∈ + 2 log (I N ), ≤ 2 log ν N⊗n U ∈ U(N )n  tr U N N dν N T (σ ) ≤ χ (σ ) (use [20, Remark 3.3] at this point). On the implying that lim T →∞ χorb orb other hand, with

L := lim

lim

T →+∞ N →∞

3 2  1 dν N ,T  U ∈ U(N ) (≤ 0), log min (U )  N2 dν N

T (σ ) ≥ χ (σ ) + n L. Hence, a similar consideration as above shows that lim T →∞ χorb orb the problem is whether L = 0 or not. We have confirmed this in the affirmative too and will give a further study on the orbital free entropy in a subsequent paper.

Acknowledgements We would like to express our sincere gratitude to the referee for his/her very careful reading of this paper and pointing out a mistake in the original proof of exponential tightness.

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References 1. Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge (2009) 2. Biane, Ph.: Free Brownian motion, free stochastic calculus and random matrices. Free Probab. Theory 12, 1–19 (1997) 3. Biane, P., Capitaine, M., Guionnet, A.: Large deviation bound for matrix Brownian motion. Invent. math. 152, 433–459 (2003) 4. Cabanal Duvillard, T., Guionnet, A.: Large deviations upper bounds for the laws of matrix-valued processes and non-commutative entropies. Ann. Probab. 29, 1205–1261 (2001) 5. Collins, B., Dahlqvist, A., Kemp, T.: The spectral edge of unitary Brownian motion. Probab. Theory Relat. Fields (2017). https://doi.org/10.1007/s00440-016-0753-x 6. Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications. Springer, Berlin (1998) 7. Guionnet, A.: Large random matrices: lectures on macroscopic asymptotics. Lecture Notes in Mathematics, vol. 1957. Springer, Berlin (2009) 8. Hiai, F., Miyamoto, T., Ueda, Y.: Orbital approach to microstate free entropy. Int. J. Math. 20, 227–273 (2009) 9. Hiai, F., Petz, D.: The Semicircle Law, Free Random Variables and Entropy. Mathematical Surveys and Monographs, vol. 77. American Mathematical Society, Providence (2000) 10. Hu, Y.: Analysis on Gaussian Spaces. World Scientific, Singapore (2016) 11. Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics), vol. 113, 2nd edn. Springer, Berlin (1998) 12. Lévy, T.: Schur–Weyl duality and the heat kernel measure on the unitary group. Adv. Math. 218, 537–575 (2008) 13. Lévy, T.: The Master Field on the Plane. Astérisque, vol. 388 (2017) 14. Lévy, T., Maïda, M.: On the Douglas–Kazakov phase transition. ESAIM Proc. 51, 89–121 (2015) 15. Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Notes Series, 335. Cambridge University Press, Cambridge (2006) 16. Nualart, D.: The Malliavin Calculus and Related Topics, Probability and its Applications, 2nd edn. Springer, Berlin (2006) 17. Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge Studies in Advanced Mathematics, 78. Cambridge University Press, Cambridge (2002) 18. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) 19. Ueda, Y.: Orbital free entropy, revisited. Indiana Univ. Math. J. 63, 551–577 (2014) 20. Ueda, Y.: A remark on orbital free entropy. Arch. Math. 108, 629–638 (2017) 21. Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104(1), 201–220 (1991) 22. Voiculescu, D.: The analogue of entropy and of Fisher’s information measure in free probability theory VI: liberation and mutual free information. Adv. Math. 146, 101–166 (1999)

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