ESI The Erwin S hr�odinger International Boltzmanngasse 9Institute for Mathemati al Physi s A-1090 Wien, AustriaRandom Walks on Symmetri Spa esand Inequalities for Matrix Spe traAlexander A. Klya hko

Vienna, Preprint ESI 900 (2000) June 20, 2000Supported by Federal Ministry of S ien e and Transport, AustriaAvailable via http://www.esi.a .at

RANDOM WALKS ON SYMMETRIC SPACESAND INEQUALITIES FOR MATRIX SPECTRAAlexander A. Klya hkoBilkent University, Ankara, TurkeyAbstra t. Using harmoni analysis on symmetri spa es we redu e the singularspe tral problem for produ ts of matri es to the re ently solved spe tral problem forsums of Hermitian matri es. This proves Thompson's onje ture [Thom℄.Introdu tionLet a point with initial position x0 in Eu lidean spa e E3 make a sequen e ofjumps x0; x1; : : : ; xn of �xed lengths ai = jxi � xi�1j in random dire tions. What an one say about the distribution of the �nal point xn?This problem has a long history partially des ribed in [Hugh℄. The �rst solutionappears in the last published paper of Lord Rayleigh [Ray℄. He dis overed thatthe probability density pn(x) is a pie ewise polynomial fun tion of the distan ed = d(x; x0) from the initial point x0 and al ulated pn expli itly for n � 6. Lateron, Treloar [Tre℄ gave a losed form of the solution for arbitrary n.In this work we apply random walks on groups and symmetri spa es (see no 2for pre ise de�nitions) to matrix spe tral problems. The main te hni al tool isa de omposition of the probability distribution by spheri al fun tions (Theorems2.3.1 and 2.4.2). We in lude a number of examples, whi h over some lassi alformulae, as well as new ones.For appli ation to the matrix spe tral problems only three examples are essential,namely the sphere S3, Eu lidean spa e E3 , and Loba hevskii spa e L3. They forma spe ial ase of a triple of symmetri spa es asso iated with any ompa t simply onne ted group G:� The group G itself;� Its Lie algebra LG;� The dual symmetri spa e HG = GC=G.For the unitary group G = SU(n) the spa e LG onsists of (skew) Hermitian tra e-less matri es, while HG = SL(n; C )=SU(n) := Hn may be identi�ed with the spa eof positive Hermitian unimodular matri es H via polar de omposition A = H � U ,A 2 SL(n; C ), U 2 SU(n). In the ase n = 2 we re over the above tripleS3 ' SU(2),E3 and L3.1991 Mathemati s Subje t Classi� ation. 15A42, , 43A90, 60J15..Key words and phrases. Eigenvalues, Spheri al fun tions, Random walks.Typeset by AMS-TEX1

2 RANDOM WALKSThe spa es G, LG andHG have positive, zero and negative urvature, and may betreated as members of one family depending on the s alar urvature �1 < K <1.Let pG, pL and pH be probability densities for random walks in G, LG and HG.For the unitary group G = SU(n) they have the following meaning:� pL(H) gives the distribution of sumsH = H1+H2+� � �+HN of independentrandom Hermitian matri es Hk with given spe tra�(Hk) = f�(k)1 � �(k)2 � � � � � �(k)n g := �(k):� pG(U ) is the distribution of produ ts U = U1U2 � � �UN of independent ran-dom unitary matri es Uk 2 SU(n) with given spe tra"(Uk) = exp(i�(k)):� pH(A) is the distribution of produ ts A = A1A2 � � �AN of random unimod-ular matri es Ak 2 SL(n; C ) with given singular spe tra�(Ak) = �(pAkA�k) = exp(�(k)):In all three ases the densities pL(H) = pL(�), pG(U ) = pG("), pH(A) = pH(�)depend only on the spe tra � = �(H), " = "(U ), � = �(A). The spe tra inturn parametrize orbits of G in the orresponding symmetri spa es. The word\random" refers to the uniform distribution in the orbits.In view of these interpretations the lassi al spe tral problems fori) sums of Hermitian matri es H1 +H2 + � � �+HN ,ii) produ ts of unitary matri es U1U2 � � �UN ,iii) singular spe trum of produ ts A1A2 � � �AN , Ak 2 SL(n; C )are just questions about the supports of the densities pL(�), pG(") and pH(�). Itturns out that the densities, and their supports, in ases i) and iii) are loselyrelated.Theorem A. Let exp : T ! T be the exponential map for a maximal torus T � Gin a ompa t simply onne ted group G, and let the previous notations be in for e.Then the following identity holds(0.1) pL(�) NYk=0Y�>0(�(k); �) = pH(exp i�) NYk=0Y�>0 sinh(�(k); �);where the internal produ t is extended over all positive roots � of G.Both sides of (0.1) are a tually polynomials in �(0) := ��; �(1); : : : ; �(N) 2 T inea h hamber de�ned by the system of hyperplanes(w0�(0) +w1�(1) + � � �+wN�(N); !i) = 0where !i are fundamental weights, and wk 2 WG are elements of Weyl group WG(Theorems 3.2.2 and 4.1.1). A similar formula holds for random walks in G, butonly for suÆ iently small � (Theorem 3.2.2).Sin e the exponential mapping for the hyperboli spa e HG is bije tive, and thedensities pL and pH di�er only by non-vanishing fa tors sinh(�(k);�)(�(k);�) , the distributionshave essentially the same supportsupp(pH) = exp(supp(pL)):For the unitary group this may be stated as follows.

A. KLYACHKO 3Theorem B. The following onditions are equivalent(1) There exist matri es Ai 2 GL(n; C ) with given singular spe tra�i = �(Ai) and � = �(A1A2 � � �AN )(2) There exist Hermitian n� n matri es Hi with spe tra�(Hi) = log�i and �(H1 +H2 + � � �+HN ) = log�:The theorem was onje tured by R. C. Thompson [Thom℄ (see also [Th-Th℄),who was inspired by the striking similarity between known results for Hermitianand singular spe tral problems. The Hermitian problem has re ently been solved bythe author [Kly℄, see [Bel, Ful, Ful2, Kn-T, Zel℄ for further improvements, in ludingHorn's onje ture. There are analogues of theorem B for orthogonal and simple ti groups.The pie ewise polynomial stru ture of the densities, whi h is given in expli itform in the last se tion of the paper, in prin iple shifts the spe tral problems intothe ombinatorial domain. Nevertheless, urrently this approa h fails to produ e asolution for the unitary spe tral problem, omparable with an elegant one given byAgnihotri and Woodward [A-W℄.Appli ation of harmoni analysis on symmetri spa es to the spe tral problemsof linear algebra was initiated by I. M. Gelfand in the early �fties. In parti ularLidski's type inequalities for the singular and the Hermitian spe tral problems were�rst proved in [G-N,B-G℄. See [D-R-W℄ for further appli ations of this approa h.The formulae for random walks in �nite groups go ba k to Frobenius [Frob℄ (upto terminology), see also [Jew, B-H℄ for related treatments in framework of hyper-groups. The main result (Theorem A) may be onsidered as a hyperboli versionof the so alled wrapping theorem for ompa t groups [D-W℄, whi h essentially isan extension of the identity (3.19) of Theorem 3.2.2 to arbitrary elements ak of Liealgebra LG. Unfortunately this extension has no probabilisti interpretation, hen eno redu tion of the unitary spe tral problem to the Hermitian one beyond region(3.20). 1. Symmetri spa es.1.1. Let's re all that a Riemann manifoldX is said to be symmetri if the geodesi symmetry � : X ! X with enter at any point x0 is an isometry. By de�nition �maps a point x on a geodesi through x0 into a symmetri point x0 on the samegeodesi and at the same distan e from x0. It follows from the de�nition that asymmetri spa e X admits a onne ted transitive Lie group of isometries G andmay be identi�ed with the homogeneous spa e X = G=K with ompa t isometrygroup K, whi h up to a �nite index may be given by one of the formulaeK = fg 2 G j gx0 = x0g = fg 2 G j g� = �gg:So in essen e symmetri spa es are parametrized by Cartan pairs (G; �) onsistingof a Lie group G and an involution � : G ! G with ompa t entralizer K. Thenthere exists an unique, up to proportionality, G-invariant metri on X = G=K andthe geodesi symmetry with enter at x0 = fKgK 7! ff��g�Kis an isometry.

4 RANDOM WALKS1.2. Examples. The following symmetri spa es are important either for motiva-tion, or for the main appli ations of our study.1.2.1. Spa es of rank one. The sphere Sn, Eu lidean spa e En , and Loba hevskiispa e Ln have evident symmetri stru tures. For example, Eu lidean spa e has Car-tan presentation En = M(n)=SO(n) with group of rigid motions M(n) as isometrygroup, and entral symmetry as Cartan involution. These are typi al examples ofspa es of rank one, for whi h double osets KnG=K depend on one parameter.1.2.2. The three spa es. A ompa t group G may be onsidered as a symmetri spa e with isometry group G � G, a ting by left and right multipli ation x 7!g1xg�12 . The Cartan involution inter hanges the fa tors in G�G, and the isotropygroup K is G itself diagonally embedded in G� G.The Lie algebra LG of a group G is a symmetri spa e with non ompa t isometrygroup generated by translations and the adjoint a tion of G.Let LGC be the omplexi� ation of LG and GC be the orresponding omplexredu tive group. Then HG = GC=G is a symmetri spa e with omplex onjugationin GC as Cartan involution. This spa e is alled the dual symmetri spa e to G.For the group SU(2) the three spa es are just the sphere S3, Eu lidean spa e E3 ,and Loba hevskii spa e L3.1.2.3. Positive Hermitian matri es. The dual spa e to the unitary group SU(n),that is Hn := SL(n; C )=SU(n), may be identi�ed with the spa e of unimodular pos-itive Hermitian matri es via the polar de omposition A = H �U , with angular partU 2 SU(n), and the positive Hermitian matrix H = pA �A� as radial omponent.The eigenvalues of H are said to be the singular values of A. This is the entralexample for our study of the singular values spe tral problem.2. Random walks.2.1. We begin with the lassi al example of random walk in Eu lidean spa e En ,whi h may be de�ned as a sequen e of random points in En(2.1) 0 = x0; x1; x2; : : :xNsu h that the di�eren es Æi = xi � xi�1 are independent and uniformly distributedin spheres of given radii ai.Treating En as the symmetri spa e G=K = M (n)=SO(n) we may identify thespheres with double osets KgK. Then the randomwalk (2.1) is given by a sequen eof elements(2.2) g1; g2; � � �gN 2 Gwhi h are independent and uniformly distributed in the double osets Xi = KgiK.The original sequen e of elements (2.2) may be re onstru ted from these data asfollows xi = g1g2 � � �giK 2 G=K = X:So we arrived at the following

A. KLYACHKO 52.1.1. De�nition. A randomwalk in the symmetri spa eX = G=K is a sequen eof random elements(2.3) xi = g1g2 � � �giK 2 G=Kwhere the gi are independent and uniformly distributed in given double osetsXi = KgiK.2.1.2. Example. Random walk in spa e Hn . As we have seen in no 1:2:3 the spa eof positive Hermitian matri es Hn is a symmetri spa e with Cartan representationHn = GL(n; C )=U (n). A double oset U (n)gU (n) � Hn in this ase onsists ofmatri es A 2 GL(n; C ) with �xed singular spe trum �(A).The matrix A, onsidered as an operator in Cn , transforms the unit sphereinto an ellipsoid with semi-axis equal to the singular values of A. Hen e one mayvisualize a random walk in Hn as a sequen e of ellipsoids in Cn obtained from theunit sphere by a su ession of dilations with given oeÆ ients �(k)1 ; �(k)2 ; : : : ; �(k)nalong randomly hosen orthogonal dire tions e(k)1 ; e(k)2 ; : : : ; e(k)n .2.1.3. Notation. For given double osets Xi = KgiK in the symmetri spa eX = G=K let(2.4) PX(x) = P (X1; X2; : : : ; XN j x)be the probability density for the distribution of the �nal element x = xN in therandom walk (2.3).In the next se tion we evaluate the densities (2.4) in terms of spheri al fun tions.2.2. Spheri al fun tions. To evaluate the densities we �rst need spheri al fun -tions on the symmetri spa e X = G=K.2.2.1. De�nition. A fun tion ' 2 L2(G=K) is said to be spheri al if '(1) = 1,and the following equation holdsZK '(xky)dk = '(x)'(y); 8x; y 2 G:Note that the equation implies bi-invarian e of spheri al fun tions'(k1xk2) = '(x); 8k1; k2 2 K:The importan e of spheri al fun tions for analysis on symmetri spa es may beseen from the following properties. Let H' � L2(G=K) be the G-invariant Hilbertsubspa e generated by the spheri al fun tion '. Then(1) G : H' is an irredu ible representation (whi h is said to be spheri al), and' 2 H' is the unique, up to proportionality, bi-invariant fun tion in H'.(2) Hen e in the ompa t ase the spa e H' is �nite dimensional.(3) H' ? H for ' 6= .(4) L2(G=K) is a dire t sum (or integral for non ompa t X = G=K) of theirredu ible representations H'.For all lassi al symmetri spa es the spheri al fun tions are expli itly known[Helg, Helg2℄.

6 RANDOM WALKS2.2.2. Example. For Eu lidean spa e En =M (n)=SO(n) spheri al fun tions de-pend only on the distan e d = jxj from the origin, and may be expressed via Besselfun tions '�(x) = 2��(� + 1) � J�(�d)(�d)� ; � = n � 22 :2.2.3. Example. For a ompa t group G, onsidered as a symmetri spa e (n01.2.2), the spheri al fun tions are just normalized hara ters '(g) = �(g)=�(1) ofirredu ible representations G : U�, and the orresponding spheri al representationof G� G is H' = U� U�.2.3. Compa t ase. Now we are in position to evaluate the probability distribu-tion for a random walk in a ompa t symmetri spa e.2.3.1. Theorem. The probability density of the random walk (2.3) in a ompa tsymmetri spa e X = G=K has the following de omposition into spheri al fun tions(2.5) P (X1; X2; : : : ; XN j x) =X' dimH' � '(x) NYi=1'(Xi);where the sum runs over all spheri al fun tions.Remark. Sin e spheri al fun tions are bi-invariant, '(gi) depends only on the dou-ble oset Xi = KgiK. This explains the notation '(Xi) = '(gi).Proof. To larify the stru ture of the proof we split it into one-move steps.Step 1. For any spheri al fun tion ' and xi 2 X the following identity holds(2.6) ZK�K�����K '(k1x1k2x2 � � �kNxN )dk1dk2 � � �dkN = '(x1)'(x2) � � �'(xN ):For n = 1 the equation follows from the de�nition of spheri al fun tionZK '(kx)dk = '(1)'(x) = '(x);and simple indu tion arguments prove it in general. �Step 2. The identity of Step 1 may be rewritten in the form(2.7) ZX '(x)P (X1; X2; : : : ; XN j x)dx = '(X1)'(X2) � � �'(XN )where Xi = Kxi.Let us onsider the mapping� : K �K � � � � �K ! Xk1 � k2 � � � �kN 7! k1x1k2x2 � � �kNxN :The fun tion '(k1x1k2x2 � � �kNxN ) is onstant on the �bers of � andP (X1; X2; : : : ; XN j x)dxis equal to the volume of the �ber ��1(dx). Hen e by Fubini's theoremZK�K�����K '(k1x1k2x2 � � �kNxN )dk1dk2 � � �dkN =ZX '(x)P (X1; X2; : : : ; XN j x)dxand the result follows. �

A. KLYACHKO 7Step 3. The density has the following de omposition into series of spheri al fun -tions P (X1; X2; : : : ; XN j x) =X' '(x)('; ')'(X1)'(X2) � � �'(XN );where (f; g) = RX f(x)g(x)dx:As with any reasonable bi-invariant fun tion, the density admits a de ompositioninto spheri al harmoni sP (X1; X2; : : : ; XN j x) =X' a''(x);with oeÆ ientsa' = 1('; ') ZX P (X1; X2; : : : ; XN jx)'(x)dx (2:7)= 1('; ')'(X1)'(X2) � � �'(XN );and the result follows. �To get the �nal formula (2.5) we have to evaluate ('; ').Step 4. The following equality holds(2.8) ('; ') = 1dimH' :This step is equivalent to evaluation of the Plan herel measure forX (see below).It may be proved as follows. Let us denote by (g)H : H' ! H' the linear operatorof the spheri al representation H' orresponding to the element g 2 G. Then theoperator ZG�K(g�1kg)Hdgdk ommutes with G and hen e by S hur's lemma is a s alar(2.9) ZG�K(g�1kg)Hdgdk = � � id:Applying this operator to the spheri al fun tion '(x) we get�'(x) = ZZK�G'(g�1kgx)dkdg = ZG '(g�1)'(gx)dg;where in the last equality we make use of the fun tional equation for spheri alfun tions (stated as De�nition 2.2.1 in our exposition). For x = 1 we get � = ('; ');and taking the tra e of (2.9) we �nally get('; ') dimH' = ZZG�K �(g�1kg)dgdk = ZK �(k)dk = 1:The last integral is equal to the multipli ity of the trivial omponent in K : H',hen e is 1. �

8 RANDOM WALKS2.3.2. Example. Random walks in S3. We identify the sphere with the groupSU(2). Then by example 2.2.3, the normalized hara ter 'k = sin k�k sin � of the ir-redu ible k-dimensional representation G : Uk is a spheri al fun tion, and Hk =Uk Uk is the orresponding spheri al representation of SU(2)� SU(2). ApplyingTheorem 2.3.1, we arrive at the formulaP (�1; �2; : : :�N j x) = 1Xk=1 1kN�1 sin k�sin � Yi sin k�isin�iwhere the random walk is de�ned by a sequen e of independent jumps by angles�1; �2; : : :�N , beginning at the North pole (� = 0), and � = �(x) is the latitude ofthe �nal point x 2 S3.Rather unexpe tedly we may sum up the series and get a �nite answer (by God'swill the wonder repeats itself in all ompa t groups). To pro eed, we �rst expresssin k� and sin k� by exponentials2�n�2i�n�1sin � sin�1 sin�2 : : : sin�nX� Xk 6=0(�1)#(�) eik(����1��2������n)kn�1 ;where the �rst sum runs over all ombinations of signs �. Then apply the Fourierexpansion for Bernoulli polynomials B�(x)Xk 6=0 e2�ikxk� = � (2�i)��! eB�(x);where eB�(x + 1) = eB�(x) and eB�(x) = B�(x) for 0 < x < 1. As result we �nallyget PS3(�1; �2; : : :�N j x) =(2.10) �n�1(n� 1)!4 sin �Qn1 sin�iX� (�1)#(�) eBn�1�� � �1 � � � � � �n2� � ;where we ex lude the �rst � sign using the symmetry eB�(�x) = (�1)� eB�(x).2.3.3. Example. Random walks in E3 . Let's now suppose that the jumps �i � 0are so small that the �nal point x never rea hes the South pole, that is(2.11) �1 + �2 + � � �+ �n < �:Then the formula (2.10) may be simpli�ed as followsPS3(�1; �2; : : :�N j x) =(2.12) �(n� 2)!2n sin �Qn1 sin�i X���1��2�����n<0(�1)#(�)(� � �1 � � � � � �n)n�2:For the proof, let's note that the sum over signs � in (2.10) is nothing but the n-thdi�eren e. Hen e for any polynomial Bn�1(x) of degree n� 1 the sum vanishesX� (�1)#(�)Bn�1�� � �1 � � � � � �n2� � = 0:

A. KLYACHKO 9The fun tion eBn�1 in (2.10) is not polynomial, but under ondition (2.11) its ar-gument spreads over two intervals of polynomiality (�1; 0) and (0; 1). Splitting thesum into two polynomial partsX���1������n>0(�1)#(�)Bn�1�� � �1 � � � � � �n2� �++ X���1������n<0(�1)#(�)Bn�1�1 + � � �1 � � � � � �n2� � ;and using the fun tional equation B�(x+ 1)� B�(x) = �x��1 we get the result.Let's now suppose that the radius of the sphere S3 tends to in�nity in su h away that R� ! d and R�i ! ai. Then taking limits in (2.12) we get the Treloarformula [Tre℄ for random walks in E3PE3(a1; a2; : : :an j d) = limR!1 12�2R3PS3(�1; �2; : : :�n j �) =(2.13) 1�(n � 2)!2n+1da1a2 � � �an Xd�a1�a2����an<0(�1)#(�)(d� a1 � a2 � � � � an)n�2;where 2�2R3 = volS3.2.4. Plan herel measure and non ompa t ase. For a non ompa t symmet-ri spa e X = G=K the spheri al representations H' are usually in�nite dimen-sional, and formula (2.5) makes no sense. Nevertheless on the spa e of spheri alfun tions (denote it by �) there exists the so- alled Plan herel measure d�(�), whi hmay be hara terized by the equation(2.14) ZG jf(g)j2dg = Z� j bf(�)j2d�(�)for any bi-invariant fun tion f 2 L2(KnG=K). Here(2.15) bf (�) = ZG f(g)'�(g)dgis the spheri al transform of f .2.4.1. Example. For a ompa t group G the Plan herel measure is dis rete. Toevaluate the measure of a spheri al fun tion f = '� we begin with its spheri altransform bf ( ) = ZG '�(g)' (g)dg = ('�; ' )Æ� ;and substitute this value in (2.14)('�; '�)2�(�) = ('�; '�):Then by (2.8) �(�) = 1('�; '�) = dimH('�):The last step in the proof of Theorem 2.3.1 is nothing but a omputation of thePlan herel measure. In a sense the Plan herel measure is an analogue of dimensionfor in�nite-dimensional spheri al representations. The Plan herel measure is knownfor all Riemannian symmetri spa es [Helg, Helg2℄.

10 RANDOM WALKS2.4.2. Theorem. The density of a random walk in an arbitrary symmetri spa eX = G=K is given by the formula(2.16) P (X1; X2; : : :XN jx) = Z� '�(x0)Yi '�(Xi)d�(�);where x0 is the symmetri element to x with respe t to (the image of) the unitelement 1 2 G, from whi h the random walk begins.Proof. The �rst two steps in the proof of Theorem 2.3.1 are valid in non ompa t ase as well. The se ond step a ording to (2.15) gives the spheri al transform bP (�)of the density P (X1; X2; : : : ; XN j x):bP (�) = '�(X1)'�(X2) � � �'�(XN ):Now the theorem follows from the inversion formula for spheri al transform(2.17) f(x) = Z�'�(x) bf (�)d�(�):�2.4.3. Remark. Theorems 2.3.1 and 2.4.2 are a tually based on two propertiesof the spheri al transform (2.15): multipli ativity with respe t to the onvolutionf � h(x) = RG f(xg)h(g�1x)dg of bi-invariant fun tions[f � g = bf � bg;and inversion formula (2.17). Both of these properties hold for any ommutativehypergroup [Jew, B-H℄. This provides a general template for su h kind of results.2.4.4. Example. For Eu lidean spa e En the spheri al fun tions and the Plan herelmeasure are given by the formulae'�(x) = 2��(� + 1)J�(�r)(�r)� ; r = jxj; � = n� 22d�(�) = 2(4�)�+1�(� + 1)�n�1d�:So a random walk in En with independent steps of length a1; a2; : : :aN has thedensity P (a1; a2; : : : ; aN j x) = onst Z 10 �n�1J�(�r)(�r)� NYi=1 J�(�ai)(�ai)� d�:For the plane E2 this amounts to Kluyver's formula [Klu℄PE2(a1; a2; : : : ; aN j x) = 12� Z 10 �J0(�jxj)J0(�a1)J0(�a2) � � �J0(�aN )d�;and for n = 3 to that of Rayleigh [Ray℄(2.18) PE3(a1; a2; : : : ; aN j x) = 12�2 Z 10 �2 sin(�r)�r NYi=1 sin(�ai)�ai d�:The general ase is due to Watson [Wat℄.

A. KLYACHKO 113. The three symmetri domains3.1. Positive Hermitian matri es. Let's begin with the symmetri spa e Hnof positive Hermitian n� n matri es. The a tion of SL(n; C )H 7! AH �At; H 2 Hn ; A 2 SL(n; C ):gives rise to the Cartan presentation Hn = SL(n; C )=SU(n). An orbit of the unitarygroup SU(n) on Hn onsists of unimodularHermitian matri esH with �xed positivespe trum �(H) whi h we write in exponential form �(H) = eS , whereS : s1 � s2 � � � � � sn;(3.1) s1 + s2 + � � �+ sn = 0:The orresponding double osetC(S) � SL(n; C )==SU(n) := SU(n)nSL(n; C )=SU(n) onsists of all matri es A 2 SLn(C ) with given singular spe trum �(A) = �(pA �At).Theorem 2.4.2, when applied to Hn , yields a distribution of the singular spe trumof produ ts A = A1A2 � � �ANof independent random fa tors Ai uniformly distributed in the spa e of matri esC(Si) with given singular spe trum �(Ai) = eSi . To get an expli it formula weneed the spheri al fun tions and the Plan herel measure for Hn . They were foundby Gelfand and Naimark in 1950 (see [Hel℄, Ch. IV, Th 5.7 for Harish-Chandra'sextension on arbitrary omplex semisimple groups). The spheri al fun tions onHn are SU-invariant and hen e depend only on the spe trum eS (3.1) of a matrixH 2 Hn . They may be written in the form(3.2) '�(S) = �2i�n(n�1)=2 1!2! � � �(n� 1)! det kei�psqkQp<q(�q � �p)Qp<q(e2sq � e2sp)where � = (�1; �2; : : : ; �n) 2 Rn. One an easily see that '� is invariant withrespe t to translations �p 7! �p + � and permutations of the omponents �p. Sothe spheri al fun tions are parametrized by the one� = ��1 � �2 � � � � � �n;�1 + �2 + � � �+ �n = 0:The Plan herel measure on � is proportional toYp<q(�q � �p)2d�where d� is Lebesgue measure on � � Rn�1.

12 RANDOM WALKS3.1.1. Example. Random walk in Loba hevskii spa e L3. Let us onsider in detailthe group SL(2; C ), whi h is lo ally isomorphi to the Lorentz group SO(3; 1).Hen e in this ase the symmetri spa e of positive unimodular Hermitian matri esH2 is a model for the Loba hevskii spa e L3 = SO(3; 1)=SO(3). Theorem 2.4.2yields the following formula for random walks in Loba hevskii spa e of urvatureradius �R with jumps of length ai(3.3) PL3(a1; a2; : : : ; aN jx) = 14�2R3 Z 1�1 �2 sin d�� sinh dYi sin ai�� sinh aid�:Here d is the distan e of x from the initial point. Putting a0 = d and leaving asidethe onstants the integral redu es to the formZRNYk=0 sin ak� d��N�1 ;and may be evaluated as follows. First of all hange the real line R to the ontourR" passing around zero by a small semi ir le in the upper halfplane, and then writedown sines via exponentials1(2i)N+1 X� (�1)#(�) ZR" ei(�a0�a1�����aN )� d��N�1 :If the sum (�a0 � a1 � � � � � aN ) is positive then the ontour may be losed by abig semi ir le in the upper halfplane, hen e by the residue theorem the integral iszero. For the negative sum one an lose the ontour in the lower halfplane, and inthis aseZR" ei(�a0�a1�����aN )� d��N�1 = �2�iRes0 ei(�a0�a1�����aN )��N�1 == � 2�i(N � 2)! [i(�a0 � a1 � � � � � aN )℄N�2:As result we get losed formulae for the integralZRNYk=0 sin ak� d��N�1 =�2N�1(N � 2)! Xa0�a1�����aN<0(�1)#(�)[a0 � a1 � � � � � aN ℄N�2;and for the density (3.3) of a random walk in Loba hevskii spa e of radius RPL3(a1; : : : ; aN jx) =(3.4) 1�R32N+1(N � 2)! sinhdQk sinh ak Xd�a1�����aN<0(�1)#(�)[d� a1 � � � � � aN ℄N�2:

A. KLYACHKO 133.1.2. Remark. The last formula for Loba hevskii spa e L3 of radius R = 1 di�ersonly by simple fa tors from those of Eu lidean spa e (2.13) and the unit sphere(2.12) PE3(a1; a2; : : : ; aN jd) = PL3(a1; a2; : : : ; aN jd) sinh dd NYk=1 sinh akak(3.5) = PS3(a1; a2; : : : ; aN jd) sindd NYk=1 sin akak ;where the se ond equality holds only in the domain of inje tivity of the exponentialmapping for the sphere a1+a2+ � � �+aN < �. The origin of this striking similaritylies in the identity(3.6) Xm>0m2 NYk=0 sin akmm sin ak = Z 10 �2 NYk=0 sin ak�� sin ak d�valid for ak � 0, su h that a1 + a2 + � � �+ aN < �. In the next se tion we extendboth (3.5) and (3.6) to an arbitrary ompa t simply onne ted group.3.2. Some identities. Let � be the root system of a simply onne ted ompa tgroup G with simple roots �1; �2; : : : ; �n and fundamental weights !1; !2; : : : ; !n:We'll use the standard notation for the half-sum of the positive roots� = 12X�>0� = !1 + !2 + : : :+ !n;and write Weyl's hara ter formula in the form(3.7) �! = �!�� ; �! = Xw2W sign(w)ew! ;where ! = e!+� is stri tly inside the Weyl hamber (e! is a dominant weight). Thesummation is over the Weyl group W = WG.We'll represent the dimension of the hara ter in a similar form(3.8) dim�! = d(!)d(�) ; d(!) = Y�>0 (!; �v):The advantage of these not quite standard notations is that the hara ter �! andits dimension may by extended to a skew-symmetri fun tion of arbitrary weight� 2 �R in the spa e spanned by the weight latti e ��w� = sign (w)��; d(w�) = sign (w)d(�);and in addition d(�) is a produ t of linear forms in �.Let now exp : T ! T be the exponential mapping for a maximal torus T � G,normalized by the ondition ker(T exp! T ) = fa 2 T j(!; a) 2Z;8! 2 �g. Then�!(exp a) = �!(exp a)��(exp a) = Pw2W e2�i(w!;a)Q�>0(e�i(�;a) � e��i(�;a)) ; a 2 T :

14 RANDOM WALKSSin e the spheri al fun tions on G are normalized hara ters'!(exp a) = d(�)�!(exp a)d(!)��(exp a) ;by theorem 2.3.1 and example 2.2.3 the random walk in G with jumps exp ak hasthe density(3.9) PG(exp a) = onstQkQ�>0 sin�(�; ak) X(!;�vi )>0d(!)2 NYk=0 �!(exp ak)d(!) ;where the onstant depends only on N , and to simplify the notations we put a0 =�a.A ording to Gelfand-Naimark and Harish-Chandra [Hel, Ch. IV, Th 5.7℄ spher-i al fun tions on the dual symmetri spa e HG = GC=G are obtained from thoseof G by the formal substitution � 7! i�, and taking the element � 2 � R in thepositive Weyl hamber instead of the integer weight ! 2 �'�(exp ia) = d(i�)��(exp a)d(�)�i�(exp a) = d(i�)d(�) Pw2W e2�i(w�;a)Q�>0(e��(�;a) � e�(�;a)) :Sin e the Plan herel measure in this ase is known d�(�) / d(�)2d�, by Theorem2.4.2 we get the density of the random walk in H = HG with steps exp iak(3.10) PH(exp ia) = onstQNk=0Q�>0 sinh�(�; ak) Z(�;�vi )>0 d(�)2 NYk=0 ��(exp ak)d(�) d�;where as before we put a0 = �a.Now we are ready to prove the analogue of identity (3.6).3.2.1. Theorem. Let ak satisfy the inequalities(3.11) j(!i; w0a0 +w1a1 + � � �+wNaN )j < 1for all fundamental weights !i and wk 2 WG. Then the following identity holds(3.12.) X(!;�vi )>0d(!)2 NYk=0 �!(exp ak)d(!) = Z(�;�vi )>0 d(�)2 NYk=0 ��(exp ak)d(�) d�The sum in (3.12) runs over integral weights inside the positive Weyl hamber,while the integral is taken over the hamber itself.3.3.2. Remark. The left-hand side of (3.12) is a periodi fun tion of ak with sim-ple roots �vi as periods, while the right side is manifestly a homogeneous fun tion.Hen e equality (3.12) an't be valid for all ak. We'll see in the next se tion that thesum in (3.12) is a polynomial fun tion of a0; a1; : : : ; aN in ea h hamber de�ned byaÆne hyperplanes(3.13) (!;w0a0 + w1a1 + � � �+ wNaN ) = p 2Z

A. KLYACHKO 15for ! 2 � and wk 2 W . The theorem implies that the integral in (3.12) is polyno-mial in ea h one de�ned by hyperplanes (3.13) passing through zero.Proof of Theorem 3.2.1. We start with the Poisson summation formula(3.14) X!2� f(!) = X̀2L bf (` )valid for any reasonable fun tion f in the spa e �R spanned by the weight latti e�. Here bf is the Fourier transformbf (q) = Z�Rf(p)e�2�i(p;q)dp;and L = ker(T exp�! T ) is the dual latti e to �. We apply (3.14) to the W -invariantfun tion f(�) = d(�)2 NYk=0 ��(exp ak)d(�)vanishing on the mirrors (�; �vi ) = 0 to get(3.15) X(!;�vi )6=0 d(!)2 NYk=0 �!(exp ak)d(!) = X̀2L Z�Re�2�i(�;`)d(�)2 NYk=0 ��(exp ak)d(�) d�:Theorem 3.2.1 just says that the sum on the right of (3.15) redu es to the �rst term` = 0. For the proof let's begin with a slightly di�erent integral(3.16) Z�Rd(�)2 e2�i(�;`)d(�) NYk=0 ��(exp ak)d(�) d�whi h by W -symmetrization may be written in the form(3.17) 1jW j Z�Rd(�)2��(exp ` )d(�) NYk=0 ��(exp ak)d(�) d�:The last integral enters into the formula (3.10) for the density PH(exp(�i` )) ofthe random walk in the hyperboli spa e HG. Sin e the set exp(iL) is dis rete inHG, the density PH(exp(�i` )), and the integrals (3.16)-(3.17) vanish identi ally for` 6= 0 and suÆ iently small steps ak. Taking derivatives of the integral (3.16) in thedire tions of all positive roots �v > 0, we kill the extra fa tor d(�) = Q�v>0(�; �v)in the denominator, and arrive to the vanishing of all terms in the right-hand sideof (3.15) with ` 6= 0. This proves the identity (3.12) for small ak.The pre ise form (3.11) of the domain, in whi h the identity holds, follows frompie ewise polynomiality of its left-hand side, whi h will be proved in the next se -tion, and homogeneity of the right-hand side. �Now we are in position to establish relations between the densities PG, PL andPH of random walks in the ompa t group G, its Lie algebra LG, and the dualsymmetri spa e HG = GC=G with steps exp ak, ak, exp iak.

16 RANDOM WALKS3.2.2. Theorem. The densities PG, PL, PH are related by the formulaePL(a) = PH (exp ia) NYk=0Y�>0 sinh�(�; ak; )�(�; ak; )(3.18) = PG(exp a) NYk=0Y�>0 sin�(�; ak)�(�; ak)(3.19)where a0 = �a and the last equality is valid under the restri tion(3.20) j(!i; w0a0 +w1a1 + � � �+wNaN )j < 1for all fundamental weights !i and wk 2 W .Proof. We have to prove only the �rst identity (3.18), sin e the se ond one followsfrom theorem 3.2.1 and the formulae (3.9)-(3.10) for the densities PG and PH .To pro eed we need a formula for the density PL. We an readily get it bytreating a random walk in the Lie algebra L with steps ak as a properly res aledwalk in HG with very small steps exp(i"ak). This leads to the following al ulationPL(a1; a2; : : : ; aN ja) =lim"!0 "dim LPH(exp i"a1; exp i"a2; : : : ; exp i"aN j exp i"a) 3:10=lim"!0 C"dim LQNk=0Q�>0 sinh�(�; "ak) Z(�;�vi )>0 d(�)2 NYk=0 ��(exp "ak)d(�) d� �7!�="=C Z(�;�vi )>0 d(�)2 NYk=0 ��(exp ak)d(�) d� lim"!0 NYk=0Y�>0 "sinh�(�; "ak) 3:10=PH(exp ia1; exp ia2; : : : ; exp iaN j exp ia) NYk=0Y�>0 sinh�(�; ak)�(�; ak) : �3.2.3. Corollary. The supports of the probability measures PL and PH for randomwalks in LG and HG with steps ak and exp iak are related by the equationsuppPH = exp(isuppPL):Proof. By (3.18) the measures di�er only by nonvanishing fa tors sinh �(�;ak)�(�;ak) . �For the unitary group SU(n) this solves the Thompson's onje ture [Thom℄.3.2.4. Theorem. Let �i, i = 1; 2; : : : ; N and � be positive spe tra. Then thefollowing statements are equivalent(1) There exist matri es Ai 2 GL(n; C ) with singular spe tra �i = �(Ai) and� = �(A1A2 � � �AN ).(2) There exist Hermitian n � n matri es Hi with spe tra �(Hi) = log�i and�(H1 +H2 + � � �+HN ) = log�.

A. KLYACHKO 17Proof. Solvability of the equations �(H1 + H2 + � � � + HN ) = log� and � =�(A1A2 � � �AN ) in (Hermitian) matri es with given (singular) spe tra means that� and log� are in the supports of the orresponding measures PH and PL. Hen ethe laim follows from the previous orollary. �3.3.5. Remark. A similar result holds for other lassi al groups, say for the sin-gular spe trum of a produ t of omplex orthogonal matri es Ai 2 SO(n; C ) and thespe trum of a sum of real symmetri n� n matri es Hi.4. Pie ewise polynomialityIn this se tion we prove pie ewise polynomiality of sums like(4.1) X(!;�vi )>0d(!)2 NYk=0 �!(exp ak)d(!) ;whi h enter in the density formula (3.9) for random walks in a ompa t groupG. Our exposition follows [Kly2℄. The summands are W -invariant fun tions,hen e we may extend the sum over all nonsingular weights d(!) 6= 0. Sin e�! =Pw2W sgn(w)ew! the problem redu es to the sums of the formXd(!)6=0 e2�i(!;a)d(!)N�1 ;for a = w0a1 + w1a2 + � � �+ wNaN , wk 2 W . In addition d(!) = Q�v>0(!; �v) isa produ t of linear forms, hen e we �nally arrive at the series(4.2) fL(xj�1; �2; : : : ; �N) = X!22�i� e(!;x)(!; �1)(!; �2) � � � (!; �N ) ;where the sum runs over those ! 2 2�i� for whi h (!; �k) 6= 0. Here �i 2 L arearbitrary elements in a latti e L, � is the dual latti e, and x 2 LR.Let us onsider aÆne hyperplanes in LRof the form H + a, a 2 L where thesubspa e H � L R is spanned by some ve tors �i. They divide L R into onne ted pie es alled hambers of the system �k.4.1.1. Theorem. The fun tion (4.2) is polynomial of degree N on ea h hamber,and its highest form doesn't depend on the hamber.4.1.2. Remark. The fun tion (4.2) is well de�ned as a distribution even if the system�k doesn't span LR. For example, an empty system of ve tors gives the Æ- fun tionof latti e L (it is just another way to write the Poisson summation formula (3.14)).4.1.3. Example. Root systems. In the ase of the density fun tion (4.1) we dealwith the system of positive roots �v, ea h taken with multipli ity N � 1. It is wellknown that any subspa e spanned by a set of roots is paraboli , i.e. spanned bya part of a basis ([Bour℄, VI.1.7 Prop 24). Su h a subspa e of odimension one< �1; �2; : : : ; b�i; : : : ; �n > is orthogonal to the fundamental weight !i. Hen e the hambers of the fun tion (4.1) are de�ned by aÆne hyperplanes (!; a) = p 2 Z,with ! onjugate to a fundamental weight, and a = w0a0+w1a1+ � � �+aNwN . The

18 RANDOM WALKSsystem of hyperplanes (!; x) = p, as opposed to the mirrors (�; x) = p, behaveshighly irregularly. Apparently neither the ombinatorial stru ture of the hambers,nor even the number of the hambers modulo translations are known.Both assertions of Theorem 4.1.1 be ome evident from the following ombinato-rial des ription of the fun tion (4.2).4.1.4. Proposition. Let us de�ne ' : RN ! L R by(4.3) ' : (t1; t2; : : : ; tN ) 7! t1�1 + t2�2 + � � �+ tN�N :Then(4.4) fL(xj�1; �2; : : : ; �N) = �mean value of ht1iht2i � � � htN ion the �ber '�1(L� x) � ;where hti = [t℄� 12 = eB1(t) is the periodi extension of the �rst Bernoulli polynomial.4.1.5. Remark. The right hand side of (4.4) should be understood in the followingway. Sin e the produ t ht1iht2i � � � htN i is periodi , the mean value may be takenover se tions of the unit ube 0 � ti � 1 by the aÆne subspa es '�1(a� x), a 2 L.Equation (4.4) implies polynomiality of fL(x) near those x for whi h the aÆnesubspa es are in general position to the unit ube, i.e. do not interse t its fa esof dimension m < n = dimLR. In other words the polynomiality fails only forx � ti1�i1 + ti2�i2 + � � �+ tim�im modL, m < n, i.e. on the walls of the hambers.Proof of proposition 4.1.4. In the following we'll understand the right-hand sideof the formula (4.2) as the Fourier expansion of a generalised fun tion. In par-ti ular fL(xj;) is the Fourier expansion of Æ-fun tion of the latti e L. With thisunderstanding we have the re urren e relation(4.5) fL(xj�1; �2; : : : ; �N ) = Z 10 �t� 12� fL(x+ t�1j�2; �3; : : : ; �N)dt;whi h may be proved as followsZ 10 �1� 12�fL(x+ t�1j�2; �3; : : : ; �N )dt =X!22�i� e(x;!)(�2; !)(�3; !) � � � (�N ; !) Z 10 �t� 12� e(!;�1)tdt =X!22�i� e(x;!)(�1; !)(�2; !) � � � (�N ; !) =fL(xj�1; �2; : : : ; �N):In this al ulation we use(4.6) Z 10 �t� 12� e(!;�1)tdt = � 0; if (!; �1) = 0,1(!;�1) ; if (!; �1) 6= 0:

A. KLYACHKO 19Applying (4.5) N times we getfL(xj�1; �2; : : : ; �N ) ==Z[0;1℄N �t1 � 12� � � ��tN � 12� fL(x+ t1�1 + � � �+ tN�N )dt1dt2 : : : dtN=� mean value of ht1iht2i � � � htN ion the �ber '�1(L � x) � :In the se ond line fL(x) = fL(xj;) is the Æ-fun tion of the latti e L. �In the density fun tion (4.1) we deal with a system of positive roots � > 0, ea htaken with multipli ity N � 1. In this ase the following version of the propositionmay be more relevant.4.1.6. Corollary. The fun tionfL(xj�m11 ; �m22 ; : : : ; �mNN ) = X!22�i� e(!;x)(!; �1)m1 (!; �2)m2 � � � (!; �N )mNis equal to the mean value of the produ t QNi=1(�1)mi+1 eBmi (ti)mi! on '�1(L � x).Here eBm is the periodi extension of m-th Bernoulli polynomial on (0; 1).Proof. To get the result one has to modify the proof of the proposition, usinginstead of (4.6) the formula(�1)�+1�! Z 10 B�(t)e(!;�1)tdt = � 0; if (!; �1) = 0,1(!;�1)� ; if (!; �1) 6= 0;whi h follows from the Fourier expansion of Bernoulli polynomials (see Example2.3.2). � Referen es[A-W℄ S. Agnihotri and C. Woodward, Eigenvalues of produ ts of unitary matri es andquantum S hubert al ulus, Math. Res. Letters 5 (1998), 817{836.[Bel℄ P. Belkale, Lo al systems on P1 for S a �nite set, Preprint (1998).[B-G℄ F. A. Berezin and I. M. Gelfand, Some remarks on the theory of spheri al fun tionson symmetri Riemann manifolds, Trudy Mosk. Mat. O-va, vol. 5, 1956, pp. 311{351(Russian); Translation: Transl. Ser. Amer. Math. So ., vol. 21, 1962, pp. 193{238.[B-H℄ W.R. Bloomand H. Heyer, Harmoni analysis of probability measures on hypergroups,Walter de Gruyter, Berlin, 1995.[Bour℄ N. Bourbaki, Lie groups and Lie algebras, Springer, New-York, 1988.[D-W℄ A. N. Dooley and N. J. Wildberger, Harmoni analysis and the global exponentialmap for ompa t Lie groups, Fun t. Anal. Appl. 27 (1993), 21{27.[D-R-W℄ A. N. Dooley, J. Repka, and N. J. Wildberger, Sums of adjoint orbits, Linear andMultilinear Algebra 36 (1993), 79{101.[Frob℄ G. Frobenius, �Uber Gruppen hara tere, Sitzungsber. der Berlin Ak. (1896), 985{1021.[Ful℄ W. Fulton,Eigenvalues of sums of Hermitian matri es (after A. Klya hko), S�eminaireBourbaki 845, June 1998, Ast�erisque 252 (1998), 255{269.[Ful2℄ W. Fulton, Eigenvalues, invariant fa tors, highest weights, and S hubert al ulus,Preprint, to appear in Bull. Amer. Math. So . (2000).[G-N℄ I. M. Gelfand and M. A. Najmark, Relation between unitary representations of the omplex unimodular group and its unitary subgroup, Izv. Akad. Nauk SSSR, Ser. mat.14 (1950), 339{360. (Russian)

20 RANDOM WALKS[Helg℄ S. Helgason, Groups and geometri analysis, A ademi Press, New York, 1984.[Helg2℄ S. Helgason, Geometri analysis on symmetri spa es, Ameri an Mathemati al So i-ety, Providen e, Rhode Island, 1994.[Hugh℄ B. D. Hughes, Random walks and random environments, Clarendon Press, Oxford,1995.[Jew℄ R. I. Jewett, Spa es with an abstra t onvolution of measures, Advan es in Math. 18(1975), no. 1, 1{101.[Klu℄ J. C. Kluyver, A lo al probability problem, Pro eedings of the Se tion of S ien es,Koninklijke A ademie van Wetes happen te Amsterdam, vol. 8, Joannes M�uller, Am-sterdam, 1906, pp. 341{351.[Kly℄ A. A. Klya hko, Stable bundles, Representation theory, and Hermitian operators,Sele ta Mathemati a, New Series 4 (1998), 419{445.[Kly2℄ A. A. Klya hko, An analogue of Frobenius formula for Lie groups, XVII All UnionAlgebr. Conf., Part I, Minsk, 1983, pp. 93-94. (Russian)[Kn-T℄ A. Knutson and T. Tao, The honey omb model of GLn(C) tensor produ ts I: Proofof the saturation onje ture, J. Amer. Math. So . 12 (1999), 1055-1090.[Ray℄ W. Rayleigh, On the problem of random vibrations and random ights in one, twoand three dimensions, Phil. Mag 37 (1919), no. 6, 321{347.[Thom℄ R. C. Thompson,Matrix spe tral inequalities, John Hopkins University, 1988.[Th-Th℄ R. C. Thompson and S. Therianos, The eigenvalues and singular values of matrixsums and produ ts. VII, Canad. Math. Bull 16 (1973), 561{569.[Tre℄ L. R. G. Treloar, The statisti al length of long- hain mole ules, Transa tions of theFaraday So iety 42 (1946), 77{82.[Wat℄ G. N. Watson, A Treatise on the Theory of Bessel Fun tions, Cambridge UniversityPress, 1944.[Zel℄ A. Zelevinsky, Littlewood-Ri hardson semigroup, MSRI Preprint (1997-044).Bilkent UniversityBilkent, 06533 Ankara TurkeyE-mail address: klya hko� fen.bilkent.edu.tr