Eigenvalues of the Laguerre Process as Non-colliding Squared Bessel Processes

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1 Eigenvalues of the Laguerre Process as Non-colliding Squared Bessel Processes Wolfgang Konig, Neil O Connell Basic Research Institute in the Mathematical Science HP Laboratories Bristol HPL-BRIMS April th, 200* Wishart and Laguerre ensembles and processes, eigenvalues as diffusions, non-colliding squared Bessel processes Let A (t) be a nxp matrix with independent standard complex Brownian entries and set M (t) = A (t)* A (t). This is a process version of the Laguerre ensemble and as such we shall refer to it as the Laguerre process. The purpose of this note is to remark that, assuming n > p-, the eigenvalues of M (t) evolve like p independent squared Bessel processes of dimension 2(n-p+ ), conditioned (in the sense of Doob) never to collide. More precisely, the function h (χ ) = i<j (χ i χ j ) is harmonic with respect to p independent squared Bessel processes of dimension 2(n-p+ ), and the eigenvalue process has the same law as the corresponding Doob h-transform. In the case where the entries of A (t) are real Brownian motions, (M (t)) t 0 is the Wishart process considered by Bru [Br9]. There it is shown that the eigenvalues of M (t) evolve according to a certain diffusion process, the generator of which is given explicitly. An interpretation in terms of non-colliding processes does not seem to be possible in this case. We also identify a class of processes (including Brownian motion, squared Bessel processes and generalised Ornstein-Uhlenbeck processes) which are all amenable to the same h-transform, and compute the corresponding transition densities and upper tail asymptotics for the first collision time. * Internal Accession Date Only Approved for External Publication Fachbereich Mathematik, MA 7-5, room MA783 Technische Universitat Berlin Strasse des 7. Juni 36 D-0623 Berlin Germany Copyright Hewlett-Packard Company 200

2 EIGENVALUES OF THE LAGUERRE PROCESS AS NON-COLLIDING SQUARED BESSEL PROCESSES Wolfgang König ;2 and Neil O'Connell BRIMS, Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS2 6QZ, United Kingdom 2 on leave from TU Berlin, Germany (March 30, 200) Abstract: Let A(t) be a n p matrix with independent standard complex Brownian entries and set M(t) =A(t) Λ A(t). This is a process version of the Laguerre ensemble and as such we shall refer to it as the Laguerre process. The purpose of this note is to remark that, assuming n > p, the eigenvalues of M(t) evolve like p independent squared Bessel processes of dimension 2(n p+), conditioned (in the sense of Doob) never to collide. More precisely, the function h(x) = Q i<j (x i xj) is harmonic with respect to p independent squared Bessel processes of dimension 2(n p + ), and the eigenvalue process has the same law as the corresponding Doob h-transform. In the case where the entries of A(t) are real Brownian motions, (M(t)) t 0 is the Wishart process considered by Bru [Br9]. There it is shown that the eigenvalues of M(t) evolve according to a certain diffusion process, the generator of which is given explicitly. An interpretation in terms of non-colliding processes does not seem to be possible in this case. We also identify a class of processes (including Brownian motion, squared Bessel processes and generalised Ornstein-Uhlenbeck processes) which are all amenable to the same h-transform, and compute the corresponding transition densities and upper tail asymptotics for the first collision time. MSC Primary 5A52, 60J65; Secondary 62E0. Keywords and phrases. Wishart and Laguerre ensembles and processes, eigenvalues as diffusions, non-colliding squared Bessel processes.

3 2 WOLFGANG KÖNIG AND NEIL O'CONNELL. Introduction Let A(t) be a n p matrix with independent standard complex Brownian entries (so that each entry of A(t) has variance 2t) and set M(t) = ja(t)j 2 = A(t) Λ A(t). We shall refer to M =(M(t)) t2[0;) as the Laguerre process. In the case p =,M is a squared Bessel process of dimension 2n, usually denoted by BESQ 2n. Let (t) =( (t);:::; p (t)) be the vector of eigenvalues of M(t), ordered decreasingly such that p (t) (t) 0. (Note that M(t) is almost surely nonnegative definite for any t 0.) The process ( (t)) t 0 is a diffusion on [0; ) p with generator given by h x H n;p =2 x i 2 i + x j i +2 n i : (.) This follows from the arguments given by Bru [Br9] for the Wishart case, with minor modifications. We remark that the Focker-Planck equation associated with ( (t)) t 0 was formally derived in [AW97]. We will assume that n > p. Our main observation is that the process ( (t)) t 0 can be identified as the h-transform of p independent squared Bessel processes of dimension 2(n p+), where the function h: [0; ) p! R is given by h(x) = i; i<j (x j x i ); x =(x ;:::;x p ) 2 [0; ) p : (.2) In other words, the process behaves like p independent BESQ 2(n p+) processes conditioned never to collide. To justify this claim, we will show that the function h given by (.2) is harmonic with respect to the generator G p;d =2 x 2 i + i (.3) of a vector of p independent BESQ d, and use standard methods to compute the generator Gp;d b of the h-transform. We obtain h p xi + x bg p;d =2 x i 2 j i + i +2 i : (.4) It is now easy to see that H n;p = Gp;2(n p+) b. section. This will be presented carefully in the next As is well-known, the function h is also harmonic with respect to the generator of p- dimensional Brownian motion. This also arises in the context of random matrices. It is a classical result, due to Dyson [Dy62], that the eigenvalues of Hermitian Brownian motion (the process-version of the Gaussian unitary ensemble) evolve like independent Brownian motions conditioned never to collide (see also [Gr00]). In Lemma 3. below we identify a class of generators for which the function h is harmonic which includes both of the above. We remark

4 LAGUERRE EIGENVALUES AND NON-COLLIDING BESSEL PROCESSES 3 that Dyson also considered unitary Brownian motion, and showed that the eigenvalues in this case behave like independent Brownian motions on the circle conditioned never to collide via the complex analogue of the function h. (For more detailed information about this process see [HW96].) In the Wishart case, where the entries of A =(A(t)) t 0 are independent standard real Brownian motions, we do not see how to give asimilar interpretation for the eigenvalue process. In this case, Bru [Br9] identified the generator of the process of eigenvalues of M(t) as h x 2 x i 2 i + x j i + n i : (.5) Note the missing factor of 2 in front of the drift term. Similar remarks apply to the Gaussian ensembles: in Dyson's work [Dy62] it turned out that, in contrast to the complex case, the process version of the Gaussian orthogonal ensemble (the real case) does not admit a representation of the eigenvalue process in terms of a system of independent particles conditioned never to collide. The interpretation of the Laguerre eigenvalue processes as h-transforms can be applied to obtain alternative derivations for the eigenvalue densities of the corresponding ensemble. As is known from the theory of random matrices (see, e.g., [Ja64]), these densities are given in the following closed form. We have P( () 2 dx) = Z p;ν Λ ( ) 2 x ν j e x j dx; x > >x p 0; (.6) i; i<j where ν = n p denotes the index of BESQ 2(n p+),andz p;ν denotes the normalisation constant. In words, () has the distribution of p independent Gamma(ν)-distributed random variables, transformed with the density h(x) 2. In Section 2 weintroduce the h-transform of (BESQ d ) Ωp and its generator, and in Section 3 we establish the harmonicity of h for a certain class of processes having independent components, which includes Brownian motion, squared Bessel processes and generalized Ornstein-Uhlenbeck processes driven by Brownian motion. Furthermore we calculate the transition densities of the transformed process started at the origin and describe the upper tail asymptotics of the first collision time of the components. 2. Non-colliding squared Bessel processes Fix p 2 N and let = ((t)) t2[0;) = ( (t);:::; p (t)) t2[0;) be a diffusion on [0; ) p whose components are independent squared Bessel processes (BESQ d ) of dimension d. In the following, the dimension d is any nonnegative number. The process has the generator G p;d given by (.3). We denote the distribution of when started at x 2 [0; ) p by P x. Note that 0 is an entrance boundary for the BESQ d. In dimensions d 2, the process stays in (0; ) p after time zero for ever, and the domain of the generator consists of the functions f such that G p;d f is continuous and bounded on [0; ) and f + (0 + ) = 0. If the dimension d is

5 4 WOLFGANG KÖNIG AND NEIL O'CONNELL smaller than two, then the components of hit zero with probability one, the boundary point 0isnon-singular. If 0isreflecting, then G p;d has the same domain as above. As follows from the more general Lemma 3. below, the function h in (.2) is harmonic with respect to G p;d, hence the h-transform of (BESQ d ) Ωp is well-defined. Let us compute its generator. Lemma 2.. The generator of the h-transform of is given by h p xi + x j bg p;d f(x) =G p;d f(x)+2 + i f(x): (2.) Proof. We have b Gp;d = G p;d + (log h; ), where (g; f) =G p;d (f g) fg p;d (g) gg p;d (f) is the so-called opérateur carré du champs (see, for example, [RY9]). Hence, bg p;d f G p;d f = G p;d (f log h) fg p;d (log h) log hg p;d (f) =2 =2 =4 + d Λ x 2 i (f log h) f@ i 2 log h log h@ i 2 f (f log h) f@ i log h log h@ i f x i 2(@ i log h)(@ i f)=4 x i f =2 x i h if h p xi + x j + i f(x): (2.2) 3. Generalisations and applications In this section, we introduce further non-colliding processes by means of h-transforms of processes with independent components. Fix p 2 N and let = ((t)) t2[0;) = ( (t);:::; p (t)) t2[0;) be a diffusion on a (possibly infinite) interval I which contains 0. By Gf(x) = 2 ff2 (x i 2 i f(x)+ μ(x i i f(x); x =(x ;:::;x p ) 2 I p : (3.) we denote the generator of, where ff 2 : I! (0; )andμ: I! R. In the following we identify a class of processes for which the function h in (.2) is harmonic. Lemma 3.. Assume that any of the following cases is satisfied: Either 2 ff2 (x) =ax + b and μ(x) = c with some a; b; c 2 R, or (in the case p > 2) 2 ff2 (x) = x 2 + ax + b and μ(x) = 2(p 2)x=3 +c for some a; b; c 2 R, or (in the case p = 2) 2 ff2 (x) is arbitrary and μ(x) constant. Then h is harmonic with respect to G, i.e., Gh 0.

6 LAGUERRE EIGENVALUES AND NON-COLLIDING BESSEL PROCESSES 5 Q Proof. Abbreviate G = G ff + G μ with obvious notation. Using the Leibniz rule i g 0 i P i g0 i Q g j,oneeasily derives that G μ h(x) = h(x) i<j G ff h(x) = h(x) μ(x i ) μ(x j ) ; i<j<k (x k x j )(x j x i ) h ff 2 (x j ) ff 2 (x k ) x j x i ff 2 (x i ) x k x j i : x k x i x k x i Hence, both G μ h and G ff h are identically zero if μ is constant and 2 ff2 a polynomial of first order. However, if p > 2 and μ(x) = cx and 2 ff2 (x) = x 2, then G μ h = c p (p )h, and 2 G ff h = p(p )(p 2)h. Hence, Gh 0 for the choice c =2(p 2)=3. Lastly, in the case 3 p =2,we have that G ff h 0 since h is a polynomial of first order in this case. Note that Lemma 3. covers in particular the cases of Brownian motion, squared Bessel processes and generalised Ornstein-Uhlenbeck processes driven by Brownian motion (see [CPY0]). As an application, we compute the transition densities of the h-transform of, started at the origin, and the upper tails of the first collision time where T = infft >0: (t) =2 W g; (3.2) W = fx =(x ;:::;x p ) 2 I p : x p > >x g: (3.3) Let p t (x; y) denote the transition density of the process ( (t)) t 0, say. We will first state a general result and later discuss the special cases of Brownian motion and BESQ d. Lemma 3.2. Assume that h is harmonic for the generator of, and assume that there is a Taylor expansion p t (x; y) p t (0;y) = f t(x) m=0 (xy) m a m (t); t 0;y 2 I; (3.4) for x in a neighborhoodofzero, where a m (t) > 0 and f t (x) > 0 satisfy lim t! a m+ (t)=a m (t) =0 and f t (0) = = lim t! f t (x). Then, for any t>0 and y 2 W, lim bp x x!0 ((t) 2 dy) =C t h(y) 2 P 0 ((t) 2 dy); (3.5) x2w where C t = Q p m=0 a m(t). Furthermore, for any x 2 W, P x T >t ο Ct h(x) E 0 h((t))lf(t) 2 W g Λ ; t!: (3.6) Proof. We are going to use the formula [KM59] P x (T >t; (t) 2 dy) = ff2s p sign(ff) p t (x i ;y ff(i) ) dy; x; y 2 W; (3.7) =

7 6 WOLFGANG KÖNIG AND NEIL O'CONNELL where S p denotes the set of permutations of ;:::;p, and sign denotes the signumofapermutation. Use (3.4) in (3.7) to obtain that P x T >t; (t) 2 dy Λ = f P 0 (t) 2 dy t (x i ) x m i i a mi (t) sign(ff) y m i : (3.8) ff(i) m ;:::;m p2n0 ff2s p Observe that ff2s p sign(ff) h y m i = det m ff(i) y j i i i;;:::;p is equal to zero if m ;:::;m p are not pairwise distinct. Hence, in (3.8), we may restrict the sum on m ;:::;m p 2 N 0 to the sum on 0» m < m 2 < < m p and an additional sum on fi 2 S p and write m fi () ;:::;m fi (p) instead of m ;:::;m p. This yields that P x T >t; (t) 2 dy m = f P 0 (t) 2 dy t (x i ) a mi (t) deth y j i m i deth i;;:::;p x j i i : i;;:::;p 0»m < <m p (3.9) (3.0) Now use that the two determinants may be written using the so-called Schur function [Ma79] as m deth x j i i = h(x)schur i;;:::;p m (x); (3.) where we abbreviated m = (m ;:::;m p ). The Schur function Schur m (x) is a certain multipolynomial in x ;:::;x p whose coefficients are nonnegative integers and may be defined combinatorially. It is homogeneous of degree m + + m p p (p ) and has the properties 2 Schur m (;:::;) = Q»i<j»p h(m) (j i); Schur (0;;:::;p ) (x) =; ( if m =(0; ;:::;p ); Schur m (0;:::;0) = 0 otherwise. (3.2) Using (3.) in (3.8), we arrive at P x T >t; (t) 2 dy P 0 (t) 2 dy = h(x)h(y) In order to derive (3.5), note that f t (x i ) 0»m < <m p Schur m (x)schur m (y) a mi (t): (3.3) bp x ((t) 2 dy) =P x (T >t; (t) 2 dy) h(y) h(x) ; (3.4) multiply (3.3) by P 0 ((t) 2 dy)h(y)=h(x) and note that lim x!0 Schur m (x) = 0 unless m = (0; ;:::;p ) in which case Schur m (x) =. Recall that f t (0) = to derive that (3.5) holds.

8 LAGUERRE EIGENVALUES AND NON-COLLIDING BESSEL PROCESSES 7 Let us derive the asymptotics of P x (T > t). integrate on y 2 W to obtain P x (T >t)=h(x) f t (x i ) 0»m < <m p Schur m (x) We multiply (3.3) by P 0 ((t) 2 dy) and Z a mi (t) P 0 ((t) 2 dy) h(y)schur m (y): W (3.5) Because of the assumption that lim t! a m+ (t)=a m (t) = 0 for any m 2 N 0, it is clear that in the limit t!only the term for m =(0; ;:::;p ) survives. Recall that lim t! f t (x) = to derive (3.6). The case of Brownian motion on I = R is a special case of Lemma 3.2 with f t (x) =e x2 =(2t) and a m (t) =t m =m!. In (3.5) we recover Weyl's formula for the joint density of the eigenvalues of the Gaussian unitary ensemble (see [Me9]). The upper tail asymptotics given by (3.6) were previously obtained in [Gr00]. Let us check that the BESQ d satisfies the assumptions of Lemma 3.2. The transition density is given [BS96] by ( y ν=2 p p t (x; y) = 2t e (x+y)=(2t) xy I x ν ; t if x>0; y ν (3.6) (2t) ν+ (ν+) e y=(2t) ; if x =0; where ν = d 2 is the index of BESQd,and denotes the Gamma function and I ν (z) = m=0 z 2 2m+ν m! (ν + m +) (3.7) is the modified Bessel function of index ν. Hence, (3.4) is satisfied with f t (x) = e x=(2t) and a m (t) = (ν +)[m! (ν + m + )] (2t) 2m. In particular C t = (ν +)p (2t) p(p ) (i) (ν + i) : (3.8) Hence, we recover (.6) from (3.5), with explicit identification of the normalisation constant. The right hand side of (3.6) is identified as follows. variable z = y=(2t) to get that h(z) (ν +) ZW p E 0 h((t))lf(t) 2 W g Λ = (2t) p 2 (p ) Use (3.6) and make the change of Λ z ν i e z i dz: (3.9) Now use Selberg's integral (see (7.6.5) in [Me9]) to finally deduce that (3.6) reads P x (T >t) ο (2t) p 2 (p ) h(x)k; t!; (3.20) where K = RW h(z)q p Λ z ν i e z i dz Q p (i) (ν + i) Λ = (ν +) (ν ++ p 2 ) p! ( 3+p 2 ) ( 3 2 ) (ν ++ p 3+j ) ( ) 2 2 : (3.2) (j) (ν + j)

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