Extreme value distributions
of noncolliding diffusion processes
Abstract
Noncolliding diffusion processes reported in the
present paper are particle systems of diffusion processes
in onedimension, 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 threedimensional 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 twomatrix 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
^{3}^{3}32000 Mathematics Subject Classification(s):
Primary 15B52, 60J60, 62G32; Secondary 82C22, 17B10.
1 Introduction
For , consider the following region in ,
(1.1) 
which is called the Weyl chamber of type in the representation theory [9]. By the KarlinMcGregor 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
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 onedimensional 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
(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
with (Proposition 13 in [15]). Here denotes the probability density of eigenvalues of random matrices in the Gaussian unitary ensemble (GUE) with variance ,
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
where
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
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]
where
, and
with
(1.4) 
In [12, 13], the process is constructed, where the probability density at at time is given by
(1.5) 
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 twomatrix 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 twomatrix 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
(1.6)  
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 ,
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 KarlinMcGregor formula as
(1.7) 
where is the transition probability density of the onedimensional absorbing Brownian motion with an absorbing wall at the origin. By the reflection principle of Brownian motion, it is given as
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 onedimensional 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
(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 threedimensional 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 threedimensional Bessel bridge. It was called the system of nonintersecting Brownian excursions in [25]. The probability density at at time is given by
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],
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
where
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]
where ,
with (1.4) and
The process is studied in [13, 18], where the probability density at at time is given by
(1.9)  
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 matrixvalued Brownian bridge such that its distribution is the same as the distribution of random matrices in the class C with variance , and a matrixvalued 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 matrixvalued 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
(1.10) 
2 Results
For , let be the th Hermite polynomial
where denotes the largest integer that is not greater than . We define the function of with an index ,
(2.1) 
If we consider the following version of Jacobi’s theta function
, where we have set and , the reciprocal relation
holds (see Sec.10.12 in [2], Sec.A.3.1 in [8]). Combining this functional equation with the fact
the following identities are derived for (Lemma 3 in [19]);
(2.2) 
The following two theorems are our main results.
Theorem 2.1
For ,
(2.3)  
and
(2.4)  
where
with
Theorem 2.2
For ,
(2.5) 
and
(2.6) 
where
with
Remarks.

(2) For , (2.5) gives
since . This is a classical result for the height distribution of the threedimensional Bessel bridge. It should be remarked that this result for distribution is equivalent with the fact on moments
with
where is the Riemann zeta function
See [3] for interesting relations of the probability laws of onedimensional conditional Brownian motions to the Jacobi theta functions and the Riemann zeta function.

(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 ,
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].
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
For , we introduce the following restricted region in ,
Then we consider the absorbing Brownian motion in this region starting from . Let the transition probability density of a onedimensional Brownian motion in an interval , where two absorbing walls are put at and , be . Then, by the KarlinMcGregor 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
The probability density