Table Of Content

A FREE ANALOGUE OF THE TRANSPORTATION COST INEQUALITY ON THE CIRCLE 5 0 FUMIO HIAI1,2 ANDDEŃES PETZ1,3 0 2 n Dedicated to Professor Andra´s Prékopa on the occasion of his 75th birthday a J 3 2 The relative entropy ] A dµ log dµ if µ ν, O S(µ,ν) := dν ≪ Z + otherwise. .  h ∞ t andtheWassersteindistanceareusefuldistancesbetweenmeasures. Forprobabilitymeasures a  m µ and ν on the Euclidean space Rn, the latter is defined as [ 1 W(µ,ν):= inf d(x,y)2dπ(x,y), 1 π∈Π(µ,ν)s 2 v ZZ 9 where d(x,y) = x y and Π(µ,ν) denotes the set of all probability measures on Rn Rn 8 with marginals µkan−d νk,2i.e., π( Rn) = µ and π(Rn ) = ν. × 3 ·× ×· 1 The transportation cost inequality (TCI) obtained by M. Talagrand [15] is 0 5 W(µ,ν) S(µ,ν), 0 ≤ / where ν is the standard Gaussian measure anpd µ is any probability measure on Rn. Re- h t cently Talagrand’s inequality and its counterpart, the logarithmic Sobolev inequality (LSI) a have received a lot of attention and they have been extended from the Euclidean spaces to m Riemannian manifolds. (Contrary to Talagrand’s inequality, the LSI gives an upper bound : v for the relative entropy.) It was shown by F. Otto and C. Villani [12] that in the Riemannian i X manifold setting the TCI follows from the LSI due to D. Bakry and M. Emery [1]. r Ontheotherhand,Ph.BianeandD.Voiculescu[3]provedthefreeanalogueofTalagrand’s a TCI for compactly supported measures on the real line. They replaced the relative entropy with the relative free entropy and the Gaussian measure with the semicircular law. Based on the method of random matrix approximation, Biane [2] proved the free LSI for measures on the real line, and we made a slight generalization of Biane and Voiculescu’s free TCI [8]. We also obtained the free TCI and LSI for measures on the unit circle using large deviation results for special unitary matrices and the differential geometry of SU(n) [8, 9]. Recently M. Ledoux [10] used the random matrix method to obtain the free analogue of thePrékopa-Leindlerinequalityontherealline. FromthistogetherwiththeHamilton-Jacobi approach, he also gave different proofs of the free LSI and TCI for measures on R. The aim of the present notes is to give a new proof of the free TCI for measures on the circle following 1Supported in part byJapan-Hungary JSPS-HASJoint Project. 2Supported in part byStrategic Information and Communications R&DPromotion Schemeof MPHPT. 3Supported in part byOTKA T032662. AMS subject classification: Primary: 46L54; secondary: 60E15, 94A17, 15A52. 1 2 F.HIAIANDD.PETZ Ledoux’s idea. In this way we do not need the large deviation technique but we establish a kind of free analogue of Prékopa-Leindler inequality on the circle. 1. The Prékopa-Leindler inequality on a Riemannian manifold Let M be a complete, connected, n-dimensional Riemannian manifold with the volume measure dx and the geodesic distance d(x,y) for x,y M. For 0< θ < 1 define ∈ Z (x,y) := z M :d(x,z) = θd(x,y), d(z,y) = (1 θ)d(x,y) , θ ∈ − n o which is the locus of points playing the role of (1 θ)x+θy. In this section we first present a − result of Cordero-Erausquin, McCann and Schmuckenschla¨ger, which is an extension of the Prékopa-Leindler inequality to the Riemannian manifold setting. Then we show that this results implies the TCI on a Riemannian manifold under some conditions (slightly stronger than the Bakry-Emery criterion). Theorem 1.1. ([5, Corollary 1.2]) Assume that Ric(M) (n 1)k holds for some k R ≥ − ∈ where Ric(M) is the Ricci curvature of M. Let f,g,h : M [0, ) be Borel measurable → ∞ functions and fix 0 < θ < 1. Assume that n−1 S (d) h(z) k f(x)1−θg(y)θ ≥ S ((1 θ)d)1−θS (θd)θ (cid:18) k − k (cid:19) holds for every x,y M, z Z (x,y) and d := d(x,y), where θ ∈ ∈ sin(√kd)/√kd if k > 0, S (d) := 1 if k = 0, k  sinh(√ kd)/√ kd if k < 0. − − Then   θ 1−θ h(x)dx f(x)dx g(x)dx . ≥ ZM (cid:18)ZM (cid:19) (cid:18)ZM (cid:19) Hereitisworthnotingaknownresult: IfRic(M) (n 1)k withk > 0,thenthediameter ≥ − of M is at most π/√k (see [4, 1.26]). Write Φ (d) := (n 1) logS (d) (1 θ)logS ((1 θ)d) θlogS (θd) . θ k k k − − − − − Let (M) denote the set of(cid:0)probability Borel measures on M. Let ν (M(cid:1)) be given by dν :=M1e−Q(x)dx with a Borel function Q :M R and a normalizatio∈nMconstant Z. Write Z → R (z;x,y) := Q(z) (1 θ)Q(x) θQ(y). θ − − − Then, the above theorem is rephrased as follows: If u,v,w : M R are Borel functions and → w(z) (1 θ)u(x)+θv(y)+R (z;x,y)+Φ (d) θ θ ≥ − holds for every x,y M, z Z (x,y) and d:= d(x,y), then θ ∈ ∈ log ew(x)dν(x) (1 θ)log eu(x)dν(x)+θlog ev(x)dν(x). (1.1) ≥ − ZM ZM ZM The following transportation cost inequality in the Riemannian setting was shown in [12] based on [1]. A FREE TCI ON THE CIRCLE: A NEW PROOF 3 Theorem 1.2. ([1] and [12]) Let ν (M) be given by dν(x) := 1e−Q(x)dx with a C2 function Q :M R. If the Bakry and∈EMmery criterion Z → Ric(M)+Hess(Q) ρI (1.2) n ≥ is satisfied with a constant ρ > 0, then 1 W(µ,ν) S(µ,ν), µ (M). ≤ ρ ∈ M r Now, we assume the following condition slightly stronger than (1.2): Ric(M) αI and Hess(Q) βI (1.3) n n ≥ ≥ for some constants α > 0, β R with α+β = ρ > 0. Our goal in this section is to prove ∈ that Theorem 1.1 implies Theorem 1.2 under the assumption (1.3). We use the celebrated variational formula (or the Monge-Kantorovich duality) for the Wasserstein distance (see [16]): ρW(µ,ν)2 = sup f(x)dµ(x) g(x)dν(x) : − (cid:26)ZM ZM ρ f,g C (M), f(x) g(y)+ d(x,y)2, x,y M (1.4) b ∈ ≤ 2 ∈ (cid:27) where ρ > 0. The variational expression for the relative entropy is also useful: S(µ,ν) = sup f(x)dµ(x) log ef(x)dν(x) :f C (M) . (1.5) b − ∈ (cid:26)ZM ZM (cid:27) Furthermore, we need the Taylor expansion of logS (d): k Lemma 1.3. ∞ logS (d) = c (kd2)j k j − j=1 X with c = 1 and c > 0 for all j 1. 1 6 j ≥ Proof. Set f(x) := log sinx; then we have x ∞ 1 2x ′ f (x) = cotx = − x x2 (mπ)2 m=1 − X because of a well-known expansion of cotx. Since ∞ ∞ ∞ 1 1 1 2 f(x) = = x2j−1, − mπ 1 x − 1+ x − (mπ)2j m=1 (cid:18) − mπ mπ(cid:19) m=1j=1 X X X we get ∞ 2(2j 1)! 1 f(2j)(0) = − − π2j m2j m=1 X so that ∞ f(2j)(0) 1 1 c = = > 0. j − (2j)! jπ2j m2j m=1 X (cid:3) 4 F.HIAIANDD.PETZ Since Ric(M) (n 1)k with k = α due to the assumption Ric(M) αI in (1.3), we ≥ − n−1 ≥ n get by Lemma 1.3 ∞ Φ (d) = (n 1) 1 (1 θ)2j+1 θ2j+1 c (kd2)j θ j − − − − − j=1 X(cid:0) (cid:1) α (n 1) 1 θ3 (1 θ)3 d2 ≤ − − − − − 6(n 1) − αθ(1 (cid:0)θ) (cid:1) = − d2. (1.6) − 2 For each x,y M let z(t) (0 t 1) be a geodesic curve joining x,y with d(x,z(t)) = ∈ ≤ ≤ td(x,y). Since the assumption Hess(Q) βI in (1.3) gives n ≥ d2 Q(z(t)) βd(x,y)2, 0 t 1, dt2 ≥ ≤ ≤ we get R (z(θ);x,y) = Q(z(θ)) θQ(z(0)) (1 θ)Q(z(1)) θ − − − βθ(1 θ) − d(x,y)2. (1.7) ≤ − 2 Hence, by (1.6) and (1.7) we have ρθ(1 θ) R (z;x,y)+Φ (d(x,y)) − d(x,y)2 θ θ ≤ − 2 for every x,y M and z Z (x,y). θ ∈ ∈ Now, let f,g C (M) be such that b ∈ ρ f(x) g(y)+ d(x,y)2, x,y M. ≤ 2 ∈ Set u:= θf, v := (1 θ)g and w := 0. Then − − (1 θ)u(x)+θv(y)+R (z;x,y)+Φ (d(x,y)) θ θ − ρ θ(1 θ) f(x) g(y) d(x,y)2 0= w(z) ≤ − − − 2 ≤ n o for every x,y M and z Z (x,y). Hence Theorem 1.1 (the rephrased version (1.1)) yields θ ∈ ∈ θ log eθf(x)dν(x)+ log e−(1−θ)g(x)dν(x) 0. 1 θ ≤ ZM − ZM Letting θ 1 gives ր log ef(x)dν(x) g(x)dν(x) 0 − ≤ ZM ZM so that f(x)dµ(x) g(x)dν(x) f(x)dµ(x) log ef(x)dν(x) S(µ,ν) − ≤ − ≤ ZM ZM ZM ZM thanks to (1.5). Finally, we apply (1.4) to obtain ρW(µ,ν)2 S(µ,ν). ≤ (cid:3) A FREE TCI ON THE CIRCLE: A NEW PROOF 5 2. Free TCI on the circle Let Q : T R be a continuous function. The weighted energy integral associated with Q → is defined by E (µ) := Σ(µ)+ Q(ζ)dµ(ζ) for µ (T), Q − T ∈M Z which admits a unique minimizer ν (T) (see [14]). Set B(Q) := E (ν ) and define Q Q Q ∈ M − the relative free entropy with respect to Q by Σ (µ):= Σ(µ)+ Q(ζ)dµ(ζ)+B(Q) for µ (T). Q − T ∈ M Z It is known ([9, Teheorem 2.1], also [7, Chap. 5]) that ΣQ(µ) is the rate function of the large deviation principle (in the scale 1/N2) for the empirical eigenvalue distribution of the special unitary random matrix e 1 dλSU(Q)(U) := exp NTr (Q(U)) dU, N ZSU(Q) − N N (cid:0) (cid:1) where dU is the Haar probability measure on the special unitary group SU(N) of order N, Q(U) for U SU(N) is defined via functional calculus and Tr is the usual trace on the N ∈ N N matrices. × The Wasserstein distance W(µ,ν) between µ,ν (T) is defined with respect to the ∈ M angular distance (i.e., the geodesic distance). The following is the free TCI for measures on T proven in [8]. The aim of this section is to re-prove this by using the method of Ledoux [10]. Theorem 2.1. ([8, Theorem 2.7]) Let Q : T R be a continuous function. If there exists a constant ρ > 1 such that Q(eit) ρt2 is con→vex on R, then −2 − 2 2 W(µ,ν ) Σ (µ), µ (T). Q Q ≤ 1+2ρ ∈ M r We introduce the relative free pressure witeh respect to Q by jQ(f):= sup f dµ ΣQ(µ) :µ (T) for f CR(T). T − ∈ M ∈ (cid:26)Z (cid:27) It is known ([10] and [6]) that e jQ(f) = EQ(νA) EQ−f(νQ−f) − 1 = lim log exp NTr (f(U)) dλSU(Q)(U). (2.1) N→∞N2 ZSU(N) N N For N N and U,V,W SU(N) write (cid:0) (cid:1) ∈ ∈ R (W;U,V) := Tr (Q(W)) (1 θ)Tr (Q(U)) θTr (Q(V)). θ,N N N N − − − ThenextlemmaisasortoffreeanalogueofPrékopa-Leindler-Ledoux inequality onthecircle. Lemma 2.2. Let f,g,h : T R be Borel functions and fix 0 < θ < 1. Assume that → Tr (h(W)) (1 θ)Tr (f(U))+θTr (g(V)) N N N ≥ − θ(1 θ) +R (W;U,V) − d(U,V)2 θ,N − 4 holds for every N N, U,V SU(N) and W Z (U,V). Then θ ∈ ∈ ∈ j (h) (1 θ)j (f)+θj (g). (2.2) Q Q Q ≥ − 6 F.HIAIANDD.PETZ Proof. Since dim(SU(N)) = N2 1 and Ric(SU(N)) = N, Φ defined for M = SU(N) − 2 θ satisfies Nθ(1 θ) Φ (d) − d2 θ ≤ − 4 thanks to (1.6). Hence, for each N N, the assumption of the lemma gives ∈ NTr (h(W)) (1 θ)NTr (f(U))+θNTr (g(V)) N N N ≥ − +NR (W;U,V)+Φ (d(U,V)) θ,N θ for every U,V SU(N) and W Z (U,V). Theorem 1.1 (the rephrased version (1.1)) can θ ∈ ∈ be applied to ν := λSU(Q), u := NTr (f()), v := NTr (g()) and w := NTr (h()); hence N N · N · N · we have log exp NTr (h(U)) dλSU(Q)(U) N N ZSU(N) (cid:0) (cid:1) (1 θ)log exp NTr (f(U)) dλSU(Q)(U) N N ≥ − ZSU(N) (cid:0) (cid:1) +θlog exp NTr (g(U)) dλSU(Q)(U), N N ZSU(N) (cid:0) (cid:1) implying the inequality (2.2) thanks to (2.1). (cid:3) The assumption of the lemma is apparently too much; so the above must not be the optimal form of the free Brunn-Minkowski inequality on T. Nevertheless, it is enough to prove Theorem 2.1. For each N N and U SU(N) set Ψ(U) := Tr (Q(U)). Using a certain regularization N ∈ ∈ technique as in [8], we may assume that Q is a harmonic function in a neighborhood of the unit disk. Then, it was shown in [8, Lemma 1.3(ii)] that the convexity assumption of Q implies Hess(Ψ) ρIN2−1. This gives as in (1.7) ≥ ρθ(1 θ) R (W;U,V) − d(U,V)2 (2.3) θ,N ≤ − 2 for every U,V SU(N) and W Z (U,V). Now, let f,g C(T) be such that θ ∈ ∈ ∈ 1+2ρ f(ζ) g(η)+ d(ζ,η)2, ζ,η T. (2.4) ≤ 4 ∈ Define the optimal matching distance on TN by N δ(ζ,η) := min d(ζ ,η )2 σ∈SNvui=1 i σ(i) uX t for ζ = (ζ ,...,ζ ),η = (η ,...,η ) TN. For U SU(N) let λ(U) := (λ (U),...,λ (U)) 1 N 1 N 1 N denotetheelementofTN consistingof∈theeigenvalu∈es ofU withmultiplicities andincounter- clockwise order. It immediately follows from (2.4) that 1+2ρ Tr (f(U)) Tr (g(V))+ δ(λ(U),λ(V))2, U,V SU(N). N N ≤ 4 ∈ Since δ(λ(U),λ(V)) d(U,V) as shown in [8, (2.11)]), this gives ≤ 1+2ρ Tr (f(U)) Tr (g(V))+ d(U,V)2, U,V SU(N). (2.5) N N ≤ 4 ∈ A FREE TCI ON THE CIRCLE: A NEW PROOF 7 Set f˜:= θf, g˜:= (1 θ)g and h˜ := 0. Then, for U,V SU(N) and W Z (U,V), by (2.3) θ − − ∈ ∈ and (2.5) we get θ(1 θ) (1 θ)Tr (f˜(U))+θTr (g˜(V))+R (W;U,V) − d(U,V)2 N N θ,N − − 4 1+2ρ θ(1 θ) Tr (f(U)) Tr (g(V)) d(U,V)2 N N ≤ − − − 4 (cid:18) (cid:19) 0 = Tr (h˜(W)). N ≤ Hence, the assumption of Lemma 2.2 is satisfied so that we have (1 θ)j (θf)+θj ( (1 θ)g) j (0) = 0. Q Q Q − − − ≤ For every µ (T), by definition of j , this implies Q ∈ M (1 θ) θfdµ Σ (µ) +θ ( (1 θ)g)dν Σ (ν ) 0 Q Q Q Q − T − T − − − ≤ (cid:18)Z (cid:19) (cid:18)Z (cid:19) so that, thanks to Σ (ν ) = 0, e e Q Q e θ fdµ f dνQ ΣQ(µ). T − T ≤ (cid:18)Z Z (cid:19) Letting θ 1 gives e ր f dµ fdν Σ (µ). Q Q T − T ≤ Z Z Using (1.4) we obtain e 1+2ρ W(µ,ν )2 Σ (µ). Q Q 2 ≤ (cid:3) e Itturnsoutthat thebound2/(1+2ρ) of ourfreeTCIon Tcannotbeimproved even if we use the Riemannian Prékopa-Leindler inequality from [5]. This suggests the best possibility of the bound. Acknowledgments. We are grateful to Professor M. Ledoux for sending us his preprint [10], to Professor Y. Ueda for suggesting us the proof of free TCI by using [5], and to Dr. A. Andai for the proof of Lemma 1.3. References [1] D. Bakry and M. Emery, Diffusion hypercontractives, in Séminaire Probabilités XIX, Lecture Notes in Math., Vol. 1123, Springer-Verlag, 1985, pp. 177–206. [2] Ph.Biane,LogarithmicSobolevinequalities,matrixmodelsandfreeentropy,ActaMath.Sinica19(2003), 1–11. [3] Ph. Biane and D. Voiculescu, A free probabilistic analogue of the Wasserstein metric on the trace-state space, Geom. Funct. Anal. 11(2001), 1125–1138. [4] J.CheegerandD.G.Ebin,ComparisonTheoremsinRiemannianGeometry,North-Holland,Amsterdam- Oxford; Elsevier, NewYork, 1975. [5] D.Cordero-Erausquin,R.J.McCannandM.Schmuckenschläger,ARiemannianinterpolationinequality à la Borell, Brascamp and Lieb, Invent. Math. 146(2001), 219–257. [6] F. Hiai, M. Mizuo and D. Petz, Free relative entropy for measures and a corresponding perturbation theory,J. Math. Soc. Japan 54(2002), 679–718. [7] F. Hiai and D. Petz, The Semicircle Law, Free Random Variables and Entropy, Mathematical Surveys and Monographs, Vol. 77, Amer. Math. Soc., Providence, 2000. [8] F. Hiai, D. Petz and Y. Ueda, Free transportation cost inequalities via random matrix approximation, Probab. Theory Related Fields130(2004), 199–221. 8 F.HIAIANDD.PETZ [9] F. Hiai, D. Petz and Y. Ueda, A free logarithmic Sobolev inequality on the circle, Canad. Math. Bull., to appear. [10] M. Ledoux, A (one-dimensional) free Brunn-Minkowski inequality, to appear in C. R. Acad. Sci. Paris Sér. I Math. [11] L.Leindler,OnacertainconverseofHölderinequality.InLinearoperatorsandapproximation,Birkhäuser, Basel, 1972, pp.182–184. [12] 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. [13] A.Prékopa,Onlogarithmicconcavemeasuresandfunctions,ActaSci.Math.(Szeged)34(1973),335–343. [14] E.B.SaffandV.Totik,Logarithmic Potentials with External Fields,Springer-Verlag,Berlin-Heidelberg- New York,1997. [15] M.Talagrand,TransportationcostforGaussianandotherproductmeasures,Geom.Funct.Anal.6(1996), 587–600. [16] C. Villani, Topics in Optimal Transportation, Grad.Studiesin Math., Vol.58, Amer.Math. Soc., Provi- dence, 2003. Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, H-1053 Budapest, Rea´ltanoda u. 13-15, Hungary