On the function Tr exp(a + itb) 1

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1 On the function Tr exp(a + itb) Mark Fannes and Denes Petz Instituut voor Theoretische Fysika K.U. Leuven, B-300 Leuven, Belgium Department for Mathematical Analysis Budapest University of Technology and Economics H-5 Budapest XI., Hungary Abstract We prove, for two free semicircularly distributed selfadjoint elements a and b in a type II von Neumann algebra with faithful trace, that the function t R 7! (exp(a + itb)) is positive denite. This shows that the Bessis- Moussa-Vilani conjecture holds for large random matrices in an asymptotic sense. Keywords and phrases: Bessis-Moussa-Vilani conjecture, Gaussian random matrix, Brown measure, positive denite function, semicircular element. 000 Mathematics Subject Classication: 5A5, 5A6, 46L54 Introduction Let H and K be selfadjoint n n matrices. It is a widely known conjecture [, 4, 8] that the function t 7! Tr e H+itK () is positive denite on R. This means that for every t ; t ; : : : ; t k R the k k matrix Tr e H+i(t u t v)k k u;v= is positive semidenite or, equivalently, that there exists a measure on R such that Tr e H+itK = e itx d(x) (t R): Published as Int. J. Math. and Math. Sci. 9(00), 389{394.

2 The function () and especially its derivatives at t = 0 dene important quantities in quantum statistical mechanics. Proving positive deniteness would lead to interesting relations among them. There have been several attempts to prove this conjecture but proofs have only been obtained under additional assumptions on H and K. A summary of such results has been collected in [3]. Several strenghtenings of the conjecture are known to fail. For example, in the light of the Lie-Trotter formula, it would be sucient to show that Tr e H=m e i(tu tv)k=m k u;v= is positive semidenite. However, this is denitely not true if n; m; k 3, see [5]. The aim of this paper is to show that the conjecture holds asymptotically for random choices of high dimensional matrices. More precisely, when the conjecture is true, then for any choice of selfadjoint n n random matrices H n and K n the function t 7! n E(Tr ehn+itkn ) () is positive denite. When H n and K n have independent Gaussian entries and their distribution is invariant under unitary conjugation, they constitute a so-called random matrix model for semicircular selfadjoint operators a and b in a type II von Neumann algebra with faithful normal tracial state. If H n and K n are independent, then Voiculescu's asymptotic freeness result about Gaussian random matrices [9] tells us that a and b are in free relation. According to asymptotic freeness () converges to t 7! (e a+itb ) (3) for every t R. Hence the conjecture implies that also the function (3) is positive denite. The goal of the present note is to prove this consequence of the conjecture. For the part of free probability theory relevant to this computation we refer to [6], however a very brief account is given below. A selfadjoint operator a is standard semicircularly distributed if (a n ) = x np 4 x dx (n N); that is, if it has the same moments as the semicircular measure. For our purpose, the free relation of two standard semicircularly distributed operators

3 a and b can be understood as a very particular rule to compute of any polynomial of a and b from the moments of a and b. However, there is a more ecient way in the setting of type II von Neumann algebras. When x is an arbitrary element in a type II von Neumann algebra with faithful normal trace, then there exists a unique probability measure x, the Brown measure [] of x, such that (g(x)) = C g(z) d x (z) (4) for any function g on C that is analytic in a domain containing the spectrum of x. (The Brown measure extends the concept of spectral multiplicity of matrices.) The fact we really use is that, under the assumption of freeness, a + itb is the so-called elliptic noncommutative distribution whose Brown measure is the two dimensional Lebesgue measure restricted to an elliptic region of the complex plane ([7], Thm. 4.9). Result We are going to prove the following Theorem. Let a and b be standard semicircular distributed selfadjoint operators in a von Neumann algebra with faithful normal trace. If a and b are in free relation with respect to, then t 7! (exp(a + itb)) is a positive denite function. Let S and S be two free semicircularly distributed selfadjoint elements with means 0 and variances > 0 and > 0. We compute (exp(s + is )) using the Brown measure. The Brown measure of S + is is the two dimensional Lebesgue measure restricted to the interior of the ellipse with axes = p + and = p +, see [7]. Normalised, it reads (dxdy) = 4 + dxdy : The expectation I(; ) := (exp(s + is )) 3

4 can now be computed using (4). I(; ) = ( + ) 4 = = x +y 0 rdr x (+) 4 dxdy exp 0 dxdy exp(x + iy) + y (+) 4 p x + + p i y + r d' exp p cos ' + p ir sin ' + + : Next, we use for natural numbers n and n 0 d' cos ' n sin ' n = (n )! (n )! n! n! (n + n )! (n +n ) : (5) Using the series expansion of the exponential and formula (5), we obtain I(; ) = X n=0 n! (n + )! ( )n : All other combinations of natural powers of cos and sin give 0 contributions. Therefore, f(t) := (exp(s + is t )) = X n=0 n! (n + )! ( t ) n : Expanding ( t ) n and reordering the terms in the series f(t) = X ( t ) k k! X j=0 j! (j + k + )! : We now use the series for the modied Bessel functions I I (z) = X m=0 m! ( + m)! z +m to get f(t) = X ( t ) k I k+ () : k! 4

5 Using the standard integral representation I k+ (x) = (k + 3=) p x for the modied Bessel functions, we nd X f(t) = p ( t ) k k! (k + 3=) k+ ds e xs ( s ) k+= ds e s ( s ) k+= : Dropping all irrelevant constants, we have to show that t 7! ds e s ( s ) = X ( s )t k (k + )! is positive denite, > 0. This function is however just a positive superposition of functions of the type with (s) 0. As sin(t) t t 7! sin((s)t) t = du e iut ; we see that the functions in (6) are positive denite. We have actually computed the Fourier transform of the function (3). A straightforward computation yields (e a+itb ) = e iut sinh( p 4 u ) du : Clearly the Fourier transform of (e a+itb ) has a compact support which coincides exactly with the (convex hull of the ) spectrum of b. Acknowledgement. This work was partly carried out during the meeting "Probability and Operator Algebras with Applications in Mathematical Physics", God, Hungary, 3 August{6 September 000 organised by the Paul Erd}os Summer Research Center of Mathematics. (6) References [] L.G. Brown, Lidski's theorem in the type II case, in Geometric methods in operator algebras (Kyoto 983), Longman Sci. Tech., Harlow, 986, {35 5

6 [] D. Bessis, P. Moussa, and M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics, J. Math. Phys. 6 (975), 38{35 [3] M. Fannes, R.F. Werner, unpublished notes, 99 [4] M. Gaudin, Sur la transformee de Laplace de tr e A consideree comme fonction de la diagonale de A, Ann. Inst. Henri Poincare A 8 (978), 43{44 [5] F. Hiai and D. Petz, unpublished notes, 994 [6] F. Hiai and D. Petz, The Semicircle Law, Free Random Variables and Entropy, Mathematical Surveys and Monographs, Vol. 77, Amer. Math. Soc., Providence, 000 [7] F. Larsen, Brown measures and R-diagonal elements in nite von Neumann algebras, Ph.D. thesis, University of Southern Denmark, 999 [8] M.L. Mehta and K. Kumar, On an integral representation of the function Tr exp(a B), J. Phys. A 9 (976), 97{06 [9] D. Voiculescu, Limit laws for random matrices and free products, Invent. Math. 04 (99), 0{0 6

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