数理解析研究所講究録 1354 巻 2004 年 10-19

ioFree logarithmic Sobolev inequalities
and free transportation cost inequalities
日合文雄 (Fumio Hiai)
東北大学情報科学研究科(Graduate School of Information Sciences, Tohoku University)

植田好道 (Yoshimichi ueda)
九州大学数理学研究院(Graduate School of Mathematics, Kyushu University)

Introduction
Since its first systematic study done by L. Gross in 1975, the logarithmic Sobolev
inequality (LSI) has been discussed by many authors in various contexts, in particular,
in close connection with the notions of hypercontractivity and spectral gap. An LSI
can be understood to compare the relative Fisher information with the relative entropy.
Among other things, we here refer to the LSI due to D. Bakry and M. Emery [1] inthe general Riemannian manifold setting. Another interesting inequality was presented
by M. Talagrand [10] in 1996, called the transportation cost inequality (TCI). A TCI
compares the (quadratic) Wasserstein distance $W(\mu, \nu)$ between probability measures, (for the definition see 52 below) with, the square root of the relative
entropy. Indeed, in [10] Talagrand proved the inequality $W(\mu, \nu)\leq\sqrt{S(\mu,\nu)}$ whenis the standard Gaussian measure on , and an exposition in the case of more generalcan be found in [8] for example. On the other hand, in [9] F. Otto and C. Villa
succeeded in discovering links between the LSI and the TCI in the Riemannian manifold
setting. This, combined with [1], implies the TCI in the same situation as Bakry andEmery’s LSI. See [7, 8, 11] for more about these classical LSI and TCI as well as related
topics.
Voiculescu’s inequality in [14, Proposition 7.9] is the first free probabilistic analog
of the LSI. Extending its single variable case (see (1.5) in \S 1), Biane obtained in [2] thefree LSI (Theorem 1.2) for measures on R. To prove this, Biane applied the classical
LSI on the Euclidean space to the related selfadjoint random matrices and used theweak convergence of their mean eigenvalue distributions. In Theorem 1.3 we show thevariant of Biane’s free LSI for measures on T. The proof is based on random matrix
approximation; we can apply Bakry and Emery’s classical LSI on the special unitaryyz special unitary randomgroup, a Riemannian manifold, to the related
matrices and pass to the scaling limit as goes to .
In [5] Biane and Voiculescu obtained the free analog of Talagrand’s TCI for compactly supported(Theorem 2.3). Their proof involves the ffee process andthe complex Burgers’ equation, and it is a realization of free probability parallel of$\mu$$\sqrt{S(\mu,\nu)}$

$\nu$

$\nu$

$\mathrm{R}^{n}$$\nu$$\mathrm{S}\mathrm{U}(\mathrm{n})$$n\mathrm{x}$$n$$\mu\in \mathcal{M}(\mathrm{R})$$\infty$not only the result itself but also the proof in [9]. The proof itself justifies the above
inequality to be the right free analog of Talagrand’s TCI. But, we can reprove Biane
and Voiculescu’s free TCI in a slightly more general setting (Theorem 2.4) by making
use of random matrix approximation. Certain
selfadjoint random matrices, i.e.,
yi selfadjoint matrices, are conprobability measures on the spaceofsidered, and the classical TCI for these measures asymptotically approaches, as goesto , to the free TCI. Furthermore, a similar method using special unitary randommatrices can work to prove the free TCI for measures on (Theorem 2.5).
The detailed version of this notes is [6].
$n\mathrm{x}n$$M_{n}^{sa}$$(\cong \mathrm{R}^{n^{2}})$$n\mathrm{x}$$n$$\infty$$\mathrm{T}$1

Free LSI for measures on$\mathbb{R}$and$\mathrm{T}$Let $M$ be a smooth complete Riemannian manifold of dimension $m$ with the volume
measure $dx$ , and let
denote the Ricci curvature tensor of $M$ . For a real-valued
function I on $M$ , the Hessian of is denoted by. The set of all Borel, the relative
probability measures on $M$ is denoted by. For ,
entropy of with respect to is denoted by, which is defined by$\mathrm{R}\mathrm{i}\mathrm{c}(M)$$C^{2}$$\Psi$$\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(\Psi)$$\mathcal{M}(M)$

$\mu$

$\mu$

$\nu\in \mathrm{A}/((M)$$S(\mu, \nu)$$\nu$$S( \mu, \nu):=\int_{M}\log\frac{d\mu}{d\nu}d\mu$.
when is absolutely continuous with respect to ; otherwise
Among huge contributions to the LSI topic, Bakry and Emery [1] gave a simple“local” criterion, the sO-called Bakry and Emery criterion, for a given measure on $M$
to satisfy an LSI.
$S(\mu, \nu):=+\mathrm{o}\mathrm{o}$

$\nu$

$\mu$

$-\Psi(x)dx$
Theorem 1.1 (Bakry and Emery [1]) Let $\in C^{2}(M)$ and set
with a normalization constant Z. Assume that the Bakry and Emery criterion$d \nu(x):=\frac{1}{Z}e$$\Psi$$\mathrm{R}\mathrm{i}\mathrm{c}(M)+\mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}(\Psi)\geq\rho I_{m}$holds with a constant $\rho>0.$ Then, for every $f\in C^{\infty}(M)$ ,$\int_{M}f^{2}\log f^{2}d\nu-(\int_{M}f^{2}d\nu)\log(\int_{M}f^{2}d\nu)\leq\frac{2}{\rho}\int_{M}||\mathit{7}f(x)||^{2}d\nu(x)$Equivalently, for every$\mu\in$$\mathrm{M}(M)$absolutely continuous with respect to$S( \mu, \nu)\leq\frac{1}{2\rho}\int_{M}||\mathrm{V}\log$ $\frac{d\mu}{d\nu}||^{2}d\mu$whenever the density $d\mu/d\nu$ is smooth on $M$ .

,

$\nu$

.

one has12

In case$M=\mathrm{R}$, we notice$S( \mu, \nu)=-S(\mu)+\int_{\mathbb{R}}\Psi(x)$$\int_{1\mathrm{R}}|$$\mathrm{r}\mathrm{d}$where $p:=$ d\mu /dx.
For each$\mu\in$$.\mathrm{M}(\mathrm{R})$$d\mu(x)+\log Z$,$\log\frac{d\mu}{d\nu}(x)|^{2}d\mu(x)=\int_{\mathrm{R}}(\frac{\phi(x)}{p(x)}+\Psi’(x))^{2}d\mu(x)$, Voiculescu [12] introduced the free entropy of,$\Sigma(\mu):=\int\int_{1\mathrm{R}^{2}}\log|x-y|d\mu(x)d\mu(y)$which is the “main component” of the free entropy$\chi(\mu)=\Sigma(\mu)+\frac{3}{4}+\frac{1}{2}\log 2\pi$Assume that$\mu\in$of$\mathrm{x}(\mathrm{m})$

$\mu$

$\mu$

introduced in [13]:

.

A4(R)
A4(R) has the density $p=$ d\mu dx (with respect to the Lebesgue$\oint$measure $dx$ ) belonging to the-spaceintroduced the ffie
fioe Fisher information ofI$(\mu)$$L^{3}$$L^{3}(\mathrm{R}):=L^{3}(\mathbb{R}dx)$. In [12] Voiculescu also$\mu$$:= \frac{4\pi^{l}}{3}\int_{\mathrm{R}}p(x)^{3}dx=4\int_{\mathrm{B}}((Hp)(x))^{2}d\mu(x)$where $Hp$ is the Hilbert transfo $rm$ of

,

$p$

.

$(Hp)(x):= \lim_{\mathrm{g}[searrow] 0}\int_{|x-t|>\epsilon}\frac{p(t)}{x-t}dt$

Let be a real-valued
function on
For each[4] introduced the relative free Fisher information
defined to be$Q$$C^{1}$$\mathbb{R}$$\Phi Q(\mu)$of$\Phi_{Q}(\mu):=4\int_{\mathbb{R}}((Hp)(x)-\frac{1}{2}Q’(x))^{2}\mathrm{d}$when has the density $p=$ d\mu /dx belonging to
is a real-valued continuous function on such that$L^{3}(\mathrm{R})$$\mu$, Biane and Speicher
relative to , and it is$\mu\in \mathcal{M}(\mathrm{R})$

$\mu$

$Q$

(1.3)

x)

; otherwise to be$\mathrm{R}$$\lim_{|x|arrow+\infty}|x|$$\exp(-\mathrm{E}Q(x))$

the weighted energy integral associated with

$=0$

$Q$

for every$\epsilon>0,$

is defined by$E_{Q}(\mu):=-\mathrm{C}(\mu)$ $+ \int_{\mathrm{R}}Q(x)d\mu(x)$for$\mu\in \mathcal{M}(\mathrm{R})$.

-op. When $Q$

$+$

13

According to a fundamental result in the theory of weighted potentials, there exists a
uniquesuch that$\mu_{Q}\in \mathcal{M}(\mathrm{R})$$E_{Q}(\mu_{Q})=$

inf$\{E_{Q}(\mu) :\mu\in \mathrm{M}(\mathrm{R})\}$,

is known to be compactly). Moreover,andis finite (hence so is
supported. The minimizer
is sometimes called the equilibrium measure associated
so that the function
with . Set $B(Q):=-E_{Q}$$E_{Q}(\mu_{Q})$$\Sigma(\mu_{Q})$$\mu_{Q}$$\mu_{Q}$$Q$$(\mu_{Q})$$\overline{\Sigma}\mathrm{Q}(\mathrm{x})$$:=-\mathrm{I}(\mu)$ $+ \int_{\mathrm{J}\mathrm{R}}Q(x)\mathrm{Q}(\mathrm{x})+B(Q)$

for

(1.4)

$\mu\in \mathcal{M}(\mathrm{R})$is nonnegative and is zero only when $\mu=\mu_{Q}$ . Following Biane and Speicher [4] andBiane [2], we call the function
the relative free entropy (or modified free entropy)of relative to . We note that the formula (1.4) resembles (1.1) while the formula[is the good rate
for(1.3) is similar to (1.2). An important point is that I
for the empirical eigenvaluefunction of the large deviation principle in the scaleselfadjoint random matrix associated with .
distribution of a certain
The following free analog of LSI was shown by Biane.
$\tilde{\Sigma}(\mu)$

$\mu$

$Q$

$(\mu)$$\mu\in$$\mathrm{A}/$$(\mathrm{R})$$1/n^{2}$$Q$$n\mathrm{x}n$Theorem 1.2 (Biane [2])$Q(x)-2gx2$ is convex on$\mathrm{R}$

Assume that is a real-valued
function on$\rho>0.$
every
Then,
with a constantfor$\mu\in$$\overline{\Sigma}_{Q}(\mu)\leq\frac{1}{2\rho}\Phi_{Q}(\mu)$In particular, when$Q(x)= \frac{\rho}{2}x2$$\tilde{\Sigma}\mathrm{Q}(\mathrm{x})$$C^{1}$$Q$$=-\mathrm{I}(\mu)$$\mathrm{R}$such that
one has$\mathrm{M}(\mathrm{R})$.

with $\rho>0,$$- \frac{3}{4}$$+ \frac{\rho}{2}\int_{\mathrm{R}}x^{2}\mathrm{Q}(\mathrm{x})-\frac{1}{2}\log 0$,

whose minimizer is the $(0, 1/\rho)$ -semicircular distribution, and$\Phi Q(\mu)=\Phi(\mu)-2\rho+\rho^{2}\int_{\mathbb{R}}x^{2}d\mu(x)$.

Hence, the free LSI of Theorem 1.2 becomes$\Sigma(\mu)\geq-\frac{1}{2\rho}\Phi(\mu)-\frac{1}{2}\log\rho+\frac{1}{4}$.

Maximizing the right-hand side over $\rho>0$ gives Voiculescu’s inequality$\chi(\mu)\geq\frac{1}{2}\log\frac{2\pi e}{\Phi(\mu)}$

.

(1.5)

14

The free entropy of$\mu\in$1$\mathrm{M}(\mathrm{T})$is similarly defined as$( \mu):=\int\int_{\mathrm{T}^{2}}\log|\zeta-\eta|d\mu(\zeta)d\mu(\eta)$.

When has the density $p=$ d\mu /d\mbox{\boldmath $\zeta$} with respect tod6/2n
belonging, the free Fisher information of was introduced into the -space[15] by$(\zeta=e^{\sqrt{-1}\theta})$$d\zeta=$$\mu$$L^{3}$$L^{3}(\mathrm{T}):=L^{3}(\mathrm{T}, d\zeta)$$\mu$$F( \mu):=\int_{\mathrm{F}}l((Hp)(\zeta))^{2}d\mu(\zeta)$

where $Hp$ is the Hilbert transform of
(Hp)

,

$p$

$(e^{\sqrt{-1}\theta}):= \lim_{\epsilon[searrow] 0}\int_{\epsilon\leq|t|<\pi}\frac{p(e^{\sqrt{-1}(\theta-t)})}{\tan(\frac{t}{2})}\frac{dt}{2\pi}$Note [15] that $F(\mu)$ is also written as$F( \mu)=\frac{1}{3}(-1+\int_{\mathrm{I}}p(()^{3}d\zeta)$

When has no such density as above, $F(\mu)$ is defined to be +0.
Let be a real-valued
function on T. As in the case of measures onA
we define the relative free Fisher information (p) by$\mu$$C^{1}$$Q$$\mu\in$$\mathrm{f}(\mathrm{T})$$\mathbb{R}$for each$\mathrm{F}_{Q}$$F_{Q}( \mu):=\int_{\mathrm{T}}((Hp)(\zeta)-Q’($(())$)^{2}d \mu(\zeta)-(\int_{\mathrm{I}}Q’(\zeta)d\mu(\zeta))^{2}$’$\mathrm{W}Q\mathrm{h}:_{\mathrm{e}\mathrm{u}\mathrm{n}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{t}\mathrm{h}_{\mathrm{d}}^{\mathrm{e}}\mathrm{d}\mathrm{e}_{\mathrm{v}\mathrm{a}\mathrm{t}}^{\mathrm{n}\mathrm{s}\mathrm{i}}\mathrm{t}\mathrm{s}_{\mathrm{e}\mathrm{o}\mathrm{f}}^{p=}\mathrm{d}7\mathrm{r}}\mathrm{a}\mathrm{s}$$\mathrm{A}_{)}^{\mathrm{b}\mathrm{e}}\theta \mathrm{J}_{\mathrm{n}}^{\mathrm{o}\mathrm{n}}\mathrm{g}^{\mathrm{i}}\mathrm{n}\mathrm{g}_{Q}^{\mathrm{t}\mathrm{o}}.,\mathrm{j}_{(e)}^{\mathrm{s}_{(\%_{\theta}^{\mathrm{o}\mathrm{t}}\mathrm{h}^{\mathrm{e}}}}4\mathrm{W}\mathrm{j}_{(e).\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{w}\mathrm{e}\mathrm{i}}^{\mathrm{s}\mathrm{e}}?_{\theta}^{\mu):=+\infty}\cdot \mathrm{H}\mathrm{e}_{\mathrm{e}}^{\mathrm{r}}\mathrm{e}$energy integral$-\mathrm{I}(\mu)$

admits a unique minimizer$B(Q):=\Sigma(\mu_{Q})-$

relative to$Q$$+ \int_{\mathrm{T}}Q(()d\mu(()$$\mu_{Q}\in$for$\mu\in$$\mathrm{M}(\mathrm{T})$

M(T), the equilibrium measure associated withand introduce the relative free entropy of$/\mathrm{T}Q(\zeta)d\mu_{Q}(\zeta)$by$\tilde{\Sigma}\mathrm{q}(\mathrm{P})$$\mu\in$$:=- \Sigma(\mu)+\int_{\mathrm{T}}Q(\zeta)d\mu(\zeta)+B(Q)$

$Q$

. Set

$\mathrm{M}(\mathrm{T})$.

M(T) is the rate function of the large deviation for the
It is known thatforempirical eigenvalue distribution of a certain
unitary (or special unitary) random$\tilde{\Sigma}_{Q}(\mu)$$\mu\in$$n\mathrm{x}n$matrix.
Our free analog of LSI for measures on$\mathrm{T}$isTheorem 1.3 ([6]) Let be a real-valued function on
is convex on with a constant $\rho>-1/2$ . Then, for every$C^{1}$$Q$$\mathrm{R}$$\overline{\Sigma}_{Q}(\mu)\mathrm{S}$$\frac{1}{1+2\rho}F_{Q}(\mu)$.

$\mathrm{T}$such that$\mu\in\lambda 4(\mathrm{T})$$Q(e^{\sqrt{-1}t})_{2}-\Delta t^{2}$

one has15

The proof is based on the procedure of random matrix approximation. Namely, theof the classical one (Theorem
free analog arises as the scaling limit in the scale
1.1) on the special unitary group. We need the convergence of the empiricaleigenvalue distribution of the random matrix not only in the mean but also in thealmost sure sense that is a consequence of the corresponding large deviation principle.
(with
We also need the exact computation of the Ricci curvature tensor oftoonrespect to the Riemannian structure associated with the usual
check the sO-called Bakry and Emery criterion in Theorem 1.1.
$1/n^{2}$$\mathrm{S}\mathrm{U}(n)$$\mathrm{S}\mathrm{U}(n)$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$Free TCI for measures on

2
Let
$\mu$

,

$\mathcal{X}$$\mathbb{R}$and$M_{n}(\mathbb{C}))$$\mathrm{T}$be a Polish space with a metric . The (quadratic) Wasserstein distance between
is defined by$d$$\nu\in \mathcal{M}(\mathcal{X})$$W( \mu, \nu):=\inf_{\pi\in\Pi(\mu,\nu)}\sqrt{\int\int_{\mathcal{X}\mathrm{x}\mathcal{X}}\frac{1}{2}d(x,y)^{2}d\pi(x,y)}$,with marginals
wheredenotes the set of all probability measures on $X$
The Wasserstein distance is sometimes
and , i.e.,$=\mu$ and$d(x,y)^{2}$.
defined with the integral of
instead of
$X$
$d(x,
y)=||x-y||$, the usual Euclidean metric,
In the typical case whereandthe celebrated TCI of Talagrand [10] is$\Pi(\mu, \nu)$$\mathrm{x}\mathcal{X}$$\pi(\mathcal{X}\mathrm{x}\cdot )=\nu.$$\pi(\prime \mathrm{x}\mathcal{X})$

$\nu$

$\mu$

$\frac{1}{2}d(x, y)^{2}$$=\mathrm{R}^{n}$$W(\mu, g_{n})\mathrm{S}$$\sqrt{S(\mu,g_{n})}$,$\mu\in \mathcal{M}(\mathrm{R}^{n})$,is the standard Gaussian measure, i.e., $dgn(x):=(2\pi)^{-n/2}e^{-||x||^{2}/2}dx(dx$means the Lebesgue measure on ). This inequality is a bit extended as follows (see
[8] .

where

$g_{n}$$\mathbb{R}^{\iota}$$)$

Theorem 2.1 Let : $W$
a constant $\rho>0.$ If
then$\Psi$$arrow \mathrm{R}$withand assume that I(x) $-2e||x||^{2}$ is convex onwith a normalization constant$\mathrm{R}^{n}$$Z_{f}$$d \nu(x):=\frac{1}{Z}e^{-\Psi(x)}dx\in \mathcal{M}(\mathrm{R}^{n}.)$$W(_{7}, \nu)\leq\sqrt{\frac{1}{\rho}S(\mu,\nu)}$,$\mu\in$$\mathrm{A}/((\mathbb{R}^{\iota})$.

In [9] Otto and Villani established the interrelation between LSI and TCI by atechnique using partial differential equations. Their result, combined with Bakry andEmery’s LSI [1] (or Theorem 1.1), implies the following TCI in a setup on Riemannian manifolds, where $M$ is an m-dimensional smooth complete Riemannian manifold
equipped with the geodesic distance (x, ) and the volume measure $dx$ .
$d$

$y$

Theorem 2.2 (Bakry and Emery [1] and Otto and Villani [9]) Let I be a real-valued$-\Psi(x)dx\in$with a normalization constantfunction on $M$ and set$C^{2}$$d \nu(x):=\frac{1}{Z}e$$\mathrm{A}/\mathrm{f}(M)$$1\mathrm{S}$Z. If the Bakry and Emery criterion$\rho>0,$ then$\mathrm{R}\mathrm{i}\mathrm{c}(M)+$Hess(?),$W(\mu, \nu)\leq\sqrt{\frac{1}{\rho}S(\mu,\nu)}$$\mathrm{u}$$\in$$\geq\rho I_{m}$holds with a constant

M (M).

On the other hand, the following free analog of Talagrand’s TCI is shown by Biane
and Voiculescu [5].
Theorem 2.3 (Biane and Voiculescu [5])For every compactly supported$\mu\in$$0,2$

is the standard semicircular measure, $i.e,$ ,

,

(2.1)

,

$W(\mu, \gamma_{0,2})\leq\sqrt{-\Sigma(\mu)+\int\frac{x^{2}}{2}d\mu(x)-\frac{3}{4}}$where )$\mathrm{M}(\mathrm{R})$$d \gamma_{0,2}(x)=\frac{1}{2\pi}\sqrt{4-x^{2}}\chi \mathrm{B}-\mathrm{a},\mathrm{a}]$$(x)$ $dx$.

In [6] we present a new proof of the above free TCI in a more general situation byusing a random matrix technique. In fact, the classical TCI on the matrix space $M$
asymptotically approaches to the free analog when the matrix size goes to 00. The
following is our free TCI for probability measures on , where the relative entropy inthe classical TCI is replaced by the relative free entropy.

:$\mathrm{R}$be a real-valued function on R.
Theorem 2.4 ([6]) Let
on with a constant $\rho>0,$ then$Q$

If

$Q(x)- \frac{\rho}{2}x^{2}$is convex$\mathrm{R}$$W(\mu, \mu_{Q})\leq\sqrt{\frac{1}{\rho}\Sigma_{Q}(\mu)\sim}$for every compactly supported$\mu\in$$\mathrm{M}(\mathrm{R})$.

isIn particular, when $Q(x)=x^{2}/2$ and so $\rho=1,$ the relative free entropy; hence Theorem 2.4 is a
the inside of the square root in (2.1) and its minimizer isgeneralization of Theorem 2.3.
Next, we consider two kinds of Wasserstein distances between probability measures). The one is the Wasserstein distance with respect to the usual metric,$|"-\eta|$ ,,T, and the other is with respect to the geodesic distance (i.e., the angular
for the former and $W(\mu, \nu)$ for the latter. Of course,
distance) on T. We write .
one has.
,,The next theorem is our free TCI for measures on comparing the Wasserstein
distance with the relative free entropy.
$\Sigma_{Q}(\mu)$$\gamma_{0,2}$$\mu$$\nu\in$$\mathrm{M}\{\mathrm{T}$$\zeta$$\eta\in$$W_{1}$$|(\mu, \nu)$$W_{\mathrm{i}\cdot 1}(\mu, \nu)\leq W(7^{\mathrm{Z}}, \nu)$$\mu$$\nu\in \mathcal{M}(\mathrm{T})$$\mathrm{T}$Theorem 2.5 ([6])
such that

Let

$Q$

be a real-valued function on T. If there exists a constantis convex on , then$Q(e^{\sqrt{-1}t})-2\mathrm{A}t^{2}$$\rho>-\mathrm{i}$$\mathrm{R}$$W_{|\cdot|}(\mu, \mu_{Q})\leq W(\mu, \mu_{Q})\leq\sqrt{\frac{2}{1+2\rho}\Sigma_{Q}(\mu)\sim}$for ever$ry\mu\in$A$\mathrm{M}\{\mathrm{T})$.

17

The special case where$W_{1}$

3

.

$|$

(7,

$Q\equiv 0$

and $\rho=0$ is$\frac{d\theta}{2\pi})\leq W(\mu,$ $\frac{d\theta}{2\pi})\leq\sqrt{-2\Sigma(\mu)}$,$\mu\in \mathcal{M}(\mathrm{T})$.

Some remarks

1. The Ricci curvature tensor ofis known to be degenerate, while that of
to be of positive constant and a straightforward computation shows that the Ricci
curvature tensor ofwith respect to the Riemannian structure associated withis$\mathrm{U}(n)$$\mathrm{S}\mathrm{U}(n)$$\mathrm{S}\mathrm{U}(n)$$\mathrm{T}\mathrm{r}_{n}$$\mathrm{R}\mathrm{i}\mathrm{c}(\mathrm{S}\mathrm{U}(n))=\frac{n}{2}I_{n^{2}-1}$.

This is the reason why we need the large deviation for the eigenvalue distribution of
special unitary random matrices instead of unitary ones.

2. A special orthogonal random matrix model can be used as well to obtain the free LSI
in Theorem 1.3 and the free TCI in Theorem 2.5. Here, note that the Ricci curvaturetensor of SO(n) is.

$\mathrm{R}\mathrm{i}\mathrm{c}(\mathrm{S}\mathrm{O}(n))=\frac{n-2}{4}I_{n(n-1)/2}$

Similarly, the free TCI in Theorem 2.4 can be shown by using a real symmetric randommatrix model as well.

3. We do not know whether the bounds
in the free LSI of Theorem 1.2 as well
as 1/(1+2p) in Theorem 1.3 are best possible or not. However, a simple computation
says that the bound.
in Theorem 1.2 cannot be smaller than
$1/2\mathrm{p}$$1/4\mathrm{p}$$1/2\mathrm{p}$4. In the case of the uniform probability measure
$W$

(,
$\mu$

$\frac{d\theta}{2\pi})\leq\sqrt{-2\Sigma(\mu)}$,$d\theta/2\pi$on , our free TCI is$\mathrm{T}$$\mu\in \mathcal{M}$ $(\mathrm{T})$,while to the authors’ best knowledge the sharpest classical TCI is
$W$

(,
$\mu$

$\frac{d\theta}{2\pi})\leq\sqrt{S(\mu,\frac{d\theta}{2\pi})}$,$\mu\in$$\mathrm{A}/\mathrm{f}(\mathrm{T})$.

Thus, it seems interesting to compare these two TCI’s. But, some concrete examples
show that these are not comparable; in fact, the ratio$\frac{-\Sigma(\mu_{k}(n))}{S(\mu_{k}(n),\frac{d\theta}{2\pi})}$can be arbitrarily small and also arbitrarily large.
5. The free LSI of Theorem 1.2 is applicable in particular for measures supported in, but we can also show a different inequality which might be a proper$\mathrm{R}^{+}=[0, \infty)$

18

instead of R. To do so, we use thefree LSI in the case where the whole space issymmetrization technique transforming measures onto symmetric ones on R.
$\mathrm{R}^{+}$$\mathbb{R}^{+}$

6. For an $N$-tuple $(X_{1}, \ldots, X_{N})$ of noncommutative selfadjoint random variables in, Voiculescu [13] introduced the free entropy-probability spacea tracial
XIX,) in the microstates approach. Furthermore, in [14] he introduced the,in the,...’) and the free entropy
free Fisher information ’(y1, . . . ’ininequality[14]
approach.
microstates-free
His$W^{*}$$\ldots$$(\mathcal{M}, \mathrm{r})$$X_{N}$$X_{N}$$\Phi$$\chi^{*}(X_{1}$,...,$X_{N})$$\chi^{*}(X_{1}$$X_{N}) \geq\frac{N}{2}\log\frac{2\pi Ne}{\Phi^{*}(X_{1},\ldots,X_{N})}$is a kind of multivariable free LSI. On the other hand, Biane and Voiculescu [5] extended) for, Xn), , ,the Wasserstein distance to the multivariable case: $W$ (( ,two $N$-tuples of noncommutative random variables. In this situation, challenging problems are to show ffee LSI and free TCI for noncommutative multivariables. In this$X_{1}$$\mathrm{Y}_{N})$$(\mathrm{Y}_{1}$$\ldots$$\ldots$

connection, the inequality$\chi^{*}(X_{1}, \ldots, X_{N})\geq\chi(X_{1}, \ldots, X_{N})$obtained by Biane, Capitane and Guionnet [3] is remarkable.

References[1] D. Bakry and M. Emery, Diffusion hypercontractives, in Seminaire Probabil ttes
$XIX$, Lecture-Notes in Math., Vol. 1123, Springer-Verlag, 1985, pp. 177-206.
[2] Ph. Biane, Logarithmic Sobolev inequalities, matrix models and free entropy, Acta
Math. Sinica 19, No. 3 (2003), 1-11.

[3] P. Biane, M. Capitaine and A. Guionnet, Large deviation bounds for matrix Brownian motion, Invent. Math. 152 (2003), 433-459.
[4] Ph. Biane and R. Speicher, Free diffusions, free entropy and free Fisher information, Ann. Inst. H. Poincare’ Probab. Statist. 37 (2001), 581-606.

[5] Ph. Biane and D. Voiculescu, A free probabilistic analogue of the Wasserstein
metric on the-state space, Geom. Fund. Anal. 11 (2001), 1125-1138.
$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}-$[6] F. Hiai, D. Petz and Y. Ueda, Inequalities related to free entropy derived from).
random matrix approximation, preprint (Archive:$\mathrm{O}\mathrm{A}\oint 0310453$[7] IVI. Ledoux, Concentration of measures and logarithmic Sobolev inequalities, In
Siminaire de Probabilitis XXXIII Lecture Notes in Math., Vol. 1709, SpringerVerlag, 1999, pp. 120-216.

13[8] M. Ledoux, The Concentration of Measure Phenomenon, Mathematical Surveys
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[9] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links
with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000), 361-400.
[10] M. Talagrand, Transportation cost for Gaussian and other product measures,
Geom. hnct. Anal 6 (1996), 587-600.
[11] C. Villani, Topics in Optimal Transportation,
Amer. Math. Soc, Providence, 2003.

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