Extreme value distributions of noncolliding diffusion processes

# Extreme value distributions of noncolliding diffusion processes

Minami IZUMI 111 Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan and Makoto KATORI 222 Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan; e-mail: katori@phys.chuo-u.ac.jp
7 July 2010
###### Abstract

Noncolliding diffusion processes reported in the present paper are -particle systems of diffusion processes in one-dimension, which are conditioned so that all particles start from the origin and never collide with each other in a finite time interval , . We consider four temporally inhomogeneous processes with duration , the noncolliding Brownian bridge, the noncolliding Brownian motion, the noncolliding three-dimensional Bessel bridge, and the noncolliding Brownian meander. Their particle distributions at each time are related to the eigenvalue distributions of random matrices in Gaussian ensembles and in some two-matrix models. Extreme values of paths in are studied for these noncolliding diffusion processes and determinantal and pfaffian representations are given for the distribution functions. The entries of the determinants and pfaffians are expressed using special functions.
KEY WORDS: Noncolliding diffusion processes, random matrix theory, extreme value distributions, determinants and pfaffians, special functions 3332000 Mathematics Subject Classification(s): Primary 15B52, 60J60, 62G32; Secondary 82C22, 17B10.

## 1 Introduction

For , consider the following region in ,

 WAN={\boldmathx=(x1,x2,…,xN)∈RN:x1

which is called the Weyl chamber of type in the representation theory [9]. By the Karlin-McGregor formula [10], the density at of an -dimensional Brownian motion at time , which starts from at time 0, and is restricted on the event that it stays in during a time interval , is given by

 fAN(t,\boldmathy|\boldmathx)=det1≤i,j≤N[p(t,yi|xj)],

where is the heat kernel, . It can be regarded as the transition probability density of the absorbing Brownian motion in from to with duration .

Set , . Then we consider the -particle system of one-dimensional Brownian motions starting from a configuration at time 0 and arriving at the configuration at time , conditioned never to collide with each other in . The probability density at the configuration at time is given by

 pAN,T(t,\boldmathx;a,b)=fAN(T−t,b|\boldmathx)fAN(t,\boldmathx|a)fAN(T,b|a). (1.2)

We will call the above mentioned process the noncolliding Brownian motion from to with duration and write it as .

We can show that the limit of (1.2) in , , is given by

 lim|a|,|b|→0pAN,T(t,\boldmathx;a,b) = pAN,T(t,\boldmathx;0,0) = qGUEN(\boldmathx;σT(t)2),t∈[0,T],\boldmathx∈WAN

with (Proposition 13 in [15]). Here denotes the probability density of eigenvalues of random matrices in the Gaussian unitary ensemble (GUE) with variance ,

 qGUEN(\boldmathx;σ2)=σ−N2(2π)N/2∏Nk=1Γ(k)e−|\boldmathx|2/(2σ2)∏1≤i

where and is the Gamma function. Note that, when , . We regard (1) as the probability density at a configuration at time of the noncolliding Brownian bridges with duration denoted by .

When we make the configuration at time be arbitrary in , we have another temporally inhomogeneous system of noncolliding Brownian motion, which is denoted by . The probability density at at time is given by

 pAN,T(t,\boldmathx;a,R)=NAN(T−t,\boldmathx)fAN(t,\boldmathx|a)NAN(T,a),t∈[0,T],

where

 NAN(s,\boldmathx)=∫WANd\boldmathyfAN(s,\boldmathy|\boldmathx),s>0,\boldmathx∈WAN,

is the probability that the absorbing Brownian motion in starting from is not yet absorbed at any boundary of the region and is surviving inside of it at time . For an even integer and an antisymmetric matrix we put

 Pf1≤i,j≤n[aij]=1(n/2)!∑σ:σ(2k−1)<σ(2k),1≤k≤n/2sgn(σ)aσ(1)σ(2)aσ(3)σ(4)⋯aσ(n−1)σ(n),

where the summation is extended over all permutations of with restriction , . This expression is known as a pfaffian (see, for example, [24]). By using the de Bruijn identity [5], which will be given as Lemma 3.1 in Section 3, we have the formula [12, 13]

 NAN(s,\boldmathx)=Pf1≤i,j≤ˆN[FAij(\boldmathx√2s)],

where

 ˆN={N,if N=even,N+1,if N=odd,

, and

 FAij(\boldmathy)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩Ψ(yj−yi),if 1≤i,j≤N,1,if 1≤i≤N,j=N+1,−1,if i=N+1,1≤j≤N,0,if i=j=N+1,

with

 Ψ(u)=2√π∫u0e−v2dv. (1.4)

In [12, 13], the process is constructed, where the probability density at at time is given by

 pAN,T(t,\boldmathx;0,R)=TN(N−1)/4t−N2/22N/2∏Nk=1Γ(k/2)e−|\boldmathx|2/(2t)∏1≤i

It has been shown that (1.5) exhibits a transition from the eigenvalue distribution of GUE to the eigenvalue distribution of another ensemble of random matrices called the Gaussian orthogonal ensemble (GOE) as [13, 14]. In other words, there establishes an interesting correspondence [12] between the temporally inhomogeneous system of noncolliding Brownian motion and the two-matrix model of Pandey and Mehta [21] in the random matrix theory [20].

The equivalence of the distribution of and the eigenvalue distribution of random matrices in GUE with variance for shown by Eq.(1), and the correspondence between and the two-matrix model of Pandey and Mehta are very useful to perform computer simulations of the conditional diffusion processes and [19]. Figure 1 shows samples of paths generated by computer simulations for these two processes.

In the present paper, we study the distributions of the extreme values defined by

 L♯(N,T) = mint∈[0,T]\boldmathX♯(t)=mint∈[0,T]X♯1(t), (1.6) R♯(N,T) = maxt∈[0,T]\boldmathX♯(t)=maxt∈[0,T]X♯N(t),

for , , and .

In this paper, we also consider the noncolliding processes “with a wall at the origin”. Instead of the Weyl chamber of type given by (1.1), we consider the Weyl chamber of type ,

 WCN={\boldmathx=(x1,x2,…,xN)∈RN:0

The density at of an -dimensional Brownian motion at time , which starts from at time 0, and is restricted on the event that it stays in during a time interval , is given by the Karlin-McGregor formula as

 fCN(t,\boldmathy|\boldmathx)=det1≤i,j≤N[p(0,∞)abs(t,yi|xj)], (1.7)

where is the transition probability density of the one-dimensional absorbing Brownian motion with an absorbing wall at the origin. By the reflection principle of Brownian motion, it is given as

 p(0,∞)abs(t,y|x) = p(t,y|x)−p(t,y|−x) = 1√2πt{e−(x−y)2/(2t)−e−(x+y)2/(2t)}

Eq.(1.7) gives the transition probability density of the absorbing Brownian motion in from to with duration .

For , , we consider the -particle system of one-dimensional Brownian motions starting from a configuration at time 0 and arriving at a configuration at time , which is conditioned so that there is no collision between any pair of particles and that all particles stay positive in a time interval . The probability density at at time is given by

 pCN,T(t,\boldmathx;a,b)=fCN(T−t,b|\boldmathx)fCN(t,\boldmathx|a)fCN(T,b|a). (1.8)

The Brownian motion conditioned to stay positive for a finite time interval is called the Brownian meander with duration [22]. So we call the -particle system, whose probability density is given by (1.8), the noncolliding Brownian meander from to with duration [16] and write it as .

The three-dimensional Bessel bridge with duration is the conditional Brownian motion such that it starts from the origin, stays positive in , and first returns to the origin at time . Then the -particle system obtained by the limit can be called the noncolliding three-dimensional Bessel bridge. It was called the system of nonintersecting Brownian excursions in [25]. The probability density at at time is given by

 pCN,T(t,\boldmathx;0,0)=qCN(% \boldmathx;σT(t)2)

with (Proposition 14 in [15]). Here denotes the probability density of positive eigenvalues of random matrices in the ensemble called the class C with variance [1],

 qCN(\boldmathx;σ2)=σ−N(2N+1)(π/2)N/2∏Nℓ=1Γ(2ℓ)e−|\boldmathx|2/(2σ2)∏1≤i

When we make the configuration at time be arbitrary in , we have another noncolliding Brownian meander, which is denoted by
, with . The probability density at at time is given by

 pCN,T(t,\boldmathx;a,R+)=NCN(T−t,\boldmathx)fCN(t,\boldmathx|a)NCN(T,a),

where

 NCN(s,\boldmathx)=∫WCNd\boldmathyfCN(s,\boldmathy|\boldmathx),s>0,\boldmathx∈WCN,

is the probability that the absorbing Brownian motion in starting from is not yet absorbed at any boundary of and is surviving inside of it at time . We can prove the formula [12, 13]

 NCN(s,\boldmathx)=Pf1≤i,j≤ˆN[FCij(\boldmathx√2s)],

where ,

 FCij(\boldmathy)=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩Ψ(yi,yj),if 1≤i,j≤N,Ψ(yi),if 1≤i≤N,j=N+1,−Ψ(yj),if i=N+1,1≤j≤N,0,if i=j=N+1,

with (1.4) and

 Ψ(u1,u2) = 2π[∫u10dv1∫u2−u1u1−u2dv2e−v21−(v1−v2)2 −∫u2u1dv1∫u1+u2u2−u1dv2e−v21−(v1−v2)2].

The process is studied in [13, 18], where the probability density at at time is given by

 pCN,T(t,\boldmathx;0,R+) (1.9) =TN2/2t−N(2N+1)/2∏Nℓ=1Γ(ℓ)e−|\boldmathx|2/(2t)∏1≤i

It has been shown that (1.9) exhibits a transition from the eigenvalue distribution of random matrices in the ensemble called the class C to the eigenvalue distribution of the class CI as [18, 15].

We can perform computer simulations of a matrix-valued Brownian bridge such that its distribution is the same as the distribution of random matrices in the class C with variance , and a matrix-valued Brownian motion, whose distribution at time changes continuously from the class C distribution to the class CI distribution of random matrices as . By numerically calculating eigenvalues of these two matrix-valued diffusion processes in , we can draw samples of paths of noncolliding diffusion particles for and , as shown in Figure 2 [19].

For , , , and , we will report the distributions of the maximum value

 H♯(N,T)=maxt∈[0,T]\boldmathY♯(t)=maxt∈[0,T]Y♯N(t). (1.10)

## 2 Results

For , let be the -th Hermite polynomial

 Hk(x)=k![k/2]∑m=0(−1)m(2x)k−2mm!(k−2m)!,

where denotes the largest integer that is not greater than . We define the function of with an index ,

 Θk(u,v)=∑n∈ZHk(un+v)e−(un+v)2. (2.1)

If we consider the following version of Jacobi’s theta function

 ϑ(x,y)=∑n∈Zqn2z2n=∑n∈Ze2π√−1xn+π√−1yn2,

, where we have set and , the reciprocal relation

 ϑ(x,y)=ϑ(xy,−1y)e−π√−1x2/y(√−1y)1/2

holds (see Sec.10.12 in [2], Sec.A.3.1 in [8]). Combining this functional equation with the fact

 e2ux−u2=∞∑k=0Hk(x)ukk!,

the following identities are derived for (Lemma 3 in [19]);

 ∑n∈Znkexp(−πη2n2+2π√−1ξη2n)=(−1)k/2ηk+12kπk/2Θk(√πη,√πξη). (2.2)

The following two theorems are our main results.

###### Theorem 2.1

For ,

 P(−ℓ

and

 P(−ℓ

where

 GAij(−ℓ√T,r√T)=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩IAij,if1≤i,j≤N,IAi,if1≤i≤N,j=N+1,−IAj,ifi=N+1,1≤j≤N,0,ifi=j=N+1,

with

 zAi(x)=Θi−1(2(ℓ+r)√2T,2ℓ√2T+x)+(−1)iΘi−1(2(ℓ+r)√2T,x) IAi=∫r/√2T−ℓ/√2TzAi(x)dx, IAij=∫−ℓ/√2T
###### Theorem 2.2

For ,

 P(H0,0(N,T)

and

 P(H0,R+(N,T)

where

 GCij(h√T)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ICij,if1≤i,j≤N,ICi,if1≤i≤N,j=N+1,−ICj,ifi=N+1,1≤j≤N,0,ifi=j=N+1,

with

 zCi(x)=Θ2i−1(2h√2T,x) ICi=∫h/√2T0zCi(x)dx,

Remarks.

(1)  By the scaling property of Brownian motion, we have the equalities in distribution

 L♯(N,T)\lx@stackreld=√TL♯(N,1),R♯(N,T)\lx@stackreld=√TR♯(N,1),

for , and

 H♯(N,T)=√TH♯(N,1)

for , . As a matter of fact, the probability distributions (2.3) and (2.4) in Theorem 2.1 are functions of and , and (2.5) and (2.6) are of , respectively.

(2)  For , (2.5) gives

 P(H0,0(1,T)

since . This is a classical result for the height distribution of the three-dimensional Bessel bridge. It should be remarked that this result for distribution is equivalent with the fact on moments

 E[(H0,0(1,T))m]=2(πT2)m/2ξ(m),m∈C

with

 ξ(m)=12m(m−1)π−m/2Γ(m2)ζ(m),

where is the Riemann zeta function

 ζ(m)=∑n∈N1nm,Rem>1.

See [3] for interesting relations of the probability laws of one-dimensional conditional Brownian motions to the Jacobi theta functions and the Riemann zeta function.

(3)  In [11], case of the formula (2.5) of Theorem 2.2 and moments calculated from it have been intensively studied, which are related to the double Dirichlet series.

(4)  The formula (2.3) of Theorem 2.1 gives the joint distribution of two extreme values and of the process . The distributions of single variable was studied in [7] and [23]. Feierl [7] also reported the distribution of the width (or “range”) defined by

 W0,0(N,T) = R0,0(N,T)−L0,0(N,T) = maxt∈[0,T](X0,0N(t)−X0,01(t)).

(5)  A part of results recently reported by Borodin et al. [4] may be stated as follows in the present notations; if is even, for each fixed ,

 maxt∈[0,T/2]1√2(1−t/T)X0,0N(t)\lx@stackreld=Y0,0N/2(T/2),

where in the RHS denotes the position of the top particle at time of the -particle system . The distribution of the top path was studied by Tracy and Widom [25].

(6)  The distribution of the “maximum height” of the process has been studied by Feierl [6] and by Schehr et al. [23] in different context and by different methods. See [19] for comparison.

## 3 Proof of Theorem 2.1

Let be the indicator function of a condition . Assume that . By definition of the process and (1.6), for

 P(−ℓ

For , we introduce the following restricted region in ,

 WAN(−ℓ,r)={\boldmathx=(x1,…,xN)∈WAN:−ℓ

Then we consider the absorbing Brownian motion in this region starting from . Let the transition probability density of a one-dimensional Brownian motion in an interval , where two absorbing walls are put at and , be . Then, by the Karlin-McGregor formula, the probability density that the absorbing Brownian motion in arrives at at time , avoided from any absorption at boundary of the region , is given by

 fA(−ℓ,r)N(T,b|a)=det1≤i,j≤N[p(−ℓ,r)abs(T,bi|aj)].

The probability density