Maximum distributions of bridges of noncolliding Brownian paths
The one-dimensional Brownian motion starting from the origin at time , conditioned to return to the origin at time and to stay positive during time interval , is called the Bessel bridge with duration 1. We consider the -particle system of such Bessel bridges conditioned never to collide with each other in , which is the continuum limit of the vicious walk model in watermelon configuration with a wall. Distributions of maximum-values of paths attained in the time interval are studied to characterize the statistics of random patterns of the repulsive paths on the spatio-temporal plane. For the outermost path, the distribution function of maximum value is exactly determined for general . We show that the present -path system of noncolliding Bessel bridges is realized as the positive-eigenvalue process of the matrix-valued Brownian bridge in the symmetry class C. Using this fact computer simulations are performed and numerical results on the -dependence of the maximum-value distributions of the inner paths are reported. The present work demonstrates that the extreme-value problems of noncolliding paths are related with the random matrix theory, representation theory of symmetry, and the number theory.
Random walk (RW) models are important in physics, chemistry, and computer sciences. They can be used effectively, when we explain basic concepts of statistical physics Rei65 , stochastic processes in physics and chemistry vKam92 , and stochastic algorithms MR95 . In particular, RWs have been used to provide simple and useful models to discuss various physical phenomena in far-from-equilibrium, such as interface dynamics BS95 ; KSOYHM05 ; YSKOM07 , polymer networks dGen79 ; dGen68 ; EG95 , wetting and melting transitions Fis84 , and so on. If we take the proper spatio-temporal continuum limit, called the diffusion scaling limit, of the RW models, the Brownian motion (BM) models are obtained. By virtue of the limit procedure, the BM models are enriched with mathematics. The following example of a conditional BM will demonstrate this statement Yor95 ; BPY01 ; KIK08 .
Consider the one-dimensional standard BM, , where and . The BM conditioned to stay positive , is called the 3-dimensional Bessel process, abbreviated as BES(3), since it is equivalent in distribution with the radial part of the 3-dimensional BM and its transition probability density is given by a special case of the modified Bessel function (see Sec.II.A below and 3.3 C in KS91 , VI.3 in RY98 , IV.34 in BS02 ). When we consider the case that it starts from the origin at time and returns to the origin at time , this conditional BES(3) is called the 3-dimensional Bessel bridge with duration 1 starting from 0, since as illustrated by Fig.1, the path drawn on the (1+1)-dimensional spatio-temporal plane seems to be a bridge over the time axis. (Note that by the scaling property between space and time of BM, no generality is lost by setting the time duration be 1.) Let , denote the 3-dimensional Bessel bridge. By symmetry, we can expect that the height of the bridge attains its maximum value with the highest probability at time . The probability density for at time is readily calculated as
(see below Eq.(15)), which gives the moments if is even and if is odd. The shape of the present bridge is, however, randomly deformed, and as we can see in Fig.1 the time , at which attains its maximum, will fluctuate around the mean time . We define
We can show that the probability density for is given as
which gives the moments
with the gamma function . Here is Riemann’s zeta function
which is an important special function in number theory BPY01 ; Yor97 . In the present paper, as multivariate generalization of the Bessel bridge KT07b , we study the -path systems of the 3-dimensional Bessel bridges with duration 1, conditioned never to collide with each other in ; , with the conditions and .
The systems of RWs with nonintersecting condition were introduced by M. E. Fisher as mathematical models to describe the wetting and the melting transitions and named vicious walk models Fis84 . They have been used not only to discuss dynamics of domain walls and melting of commensurate surfaces HF84 , but also to study polymer networks dGen68 ; EG95 , the related enumerative combinatorial problems AME91 ; GOV98 ; KGV00 , and nonequilibrium critical phenomena CK03 . In general the noncolliding diffusion particle systems are obtained as the diffusion scaling limits of the vicious RW models KT02 ; KT03 ; Gill03 . In particular, the version of the vicious RW model, whose continuum limit is the noncolliding Bessel bridges, , is called the N-watermelons with a wall EG95 ; GOV98 ; KGV00 ; Gill03 ; Kra06 .
In the discrete mathematics the one-dimensional RWs conditioned to visit only nonnegative sites is called the Dyck paths and the asymptotics of the average height of the Dyck paths in the long-step limit was studied by de Bruijn, Knuth and Rice dBKR72 . Recently Fulmek generalized this classical result by evaluating the asymptotics of the average height of the 2-watermelons with a wall Ful07 . In this calculation, he showed the fact that the number-theoretical special functions, such as Jacobi’s theta function and the double Dirichlet series are useful to describe the asymptotics. Motivated by this important observation, the present authors KIK08 studied the case of the noncolliding Bessel bridges, , which is the continuum limit of the 2-watermelons with a wall, and clarified that this phenomenon is indeed the generalization of the relationship between the maximum-value distribution of the 3-dimensional Bessel bridge and Riemann’s zeta function mentioned above.
We will report in this paper both of the exact results and the numerical results on the maximum-value distributions of paths in the noncolliding Bessel bridges. Main exact result is the following determinantal expression for the maximum-value distribution of the outermost path (i.e. the height of the continuum limit of watermelons),
for the -path system ;
where denotes the -th Hermite polynomial defined by
with = the largest integer that is not greater than . (We can see that, since , Eq.(4) with and the relation give the result (2).) The long-step asymptotics of all moments of the height of the -watermelons with a wall have been fully studied for arbitrary by Feierl Fei07 ; Fei08a . We will show that our result (4) for the distribution functions of the continuous model is consistent with the results by Feierl for the moments of his discrete model. Quite recently Schehr et al. SMCRF08 showed their study on the same problem and others by the path-integral technique, a different method both from ours and Feierl’s. They also reported an expression for the maximum-value distribution (Eq.(5) in their paper SMCRF08 ), which is different from (4). We will show that the equivalence of these two expressions is guaranteed by the functional equation satisfied by Jacobi’s theta function .
Dyson introduced a matrix-valued BM, , in the space of Hermitian matrices. If the matrix size is , the diagonal elements, , the real parts of the upper-triangle elements, , and the imaginary parts of them, , are given by independent one-dimensional standard BMs, the total number of which is . Dyson showed that the eigenvalues of behaves as an interacting diffusion particle system on the real axis , in which the long-ranged repulsive forces work between any pair of particles with the strength proportional to the inverse of the distance of two particles Dys62 . This eigenvalue process is called Dyson’s BM model and it has been proved to be equivalent in distribution with the system of -BMs conditioned never to collide with each other Bia95 ; Gra99 . The correspondence between eigenvalue processes of matrix-valued diffusion processes and noncolliding particle systems has been studied Gra99 ; Bru89 ; Bru91 ; KO01 ; KT04 . In the present paper we will use the fact that the -noncolliding Bessel bridges can be realized as the positive-eigenvalue process of the matrix-valued Brownian bridge, whose distribution at each time is related with the random matrix theory Meh04 with a special symmetry called the class C in AZ96 ; AZ97 . Figure 2 shows a sample of paths of the noncolliding Bessel bridges realized by this eigenvalue process. One can imagine that it is very hard to simulate such paths all starting from the origin and returning to the origin with noncolliding condition by direct computer simulation. The present paper will demonstrate that the relationship between the random matrix ensembles and the noncolliding particle systems KT04 provides a practical method to study such conditional processes effectively by computer simulations. We will report the numerical evaluation of the maximum-value distributions of not only the outermost path , but also inner paths .
Both from the view-points of statistical physics and of random matrix theory Meh04 ; KT07b , the study of limit is interesting and important for noncolliding paths KT07a ; TW07 ; Kuij08 . For the average value of the maximum values of the outermost path, we can read the following behavior from the numerical work by Bonichon and Mosbah for the watermelons with a wall BM03 ,
In the present paper, we report the numerical study of the -dependence of the maximum-value distributions systematically not only for the outermost path but also for all inner paths and discuss the asymptotics based on our numerical data.
The paper is organized as follows. In Sec.II.A, after giving brief explanations of the 3-dimensional Bessel bridge and the Karlin-McGregor formula KM59 , we define the -noncolliding Bessel bridges by giving the transition probability densities. A matrix representation of the symmetry class C is shown in Sec.II.B and the matrix-valued Brownian bridge in the symmetry class C is introduced. The equivalence in distribution between its positive-eigenvalue process and the noncolliding Bessel bridges is then stated. The problems studied in this paper is announced in Sec.II.C. In Sec.III.A the exact expression of the distribution function of the maximum value for the outermost path, which is the same as Eq.(5) in SMCRF08 , is derived by our method (Lemma 1). This exact expression is then transformed into two kinds of determinantal expressions (Proposition 2 and Theorem 4) in Sec.III.B, one of which is Eq.(7) given above. The key lemma in the transformation is a set of equalities between infinite series involving the Hermite polynomials (Lemma 3) derived from the functional equation of Jacobi’s theta function . The numerical study is reported in Sec.IV. Concluding remarks are given in Sec.V. Appendices are used to give proofs of some formulas.
Ii Models and Problems
ii.1 Transition probability density of noncolliding Bessel bridges
Let , be the one-dimensional standard BM starting from the origin; . For any series of times , the probability that the BM stays in interval at each time , is given by
where the transition probability density from the position to during time period is given by the probability density of the normal distribution with mean 0 and variance ,
We consider the situation that an absorbing wall is set at the origin and BM is absorbed if it arrives at the wall. By the reflection principle of BM KS91 , the transition probability density of such an absorbing BM is given by
for . The survival probability of the absorbing BM starting from for the time period is then given by
whose asymptotics in is easily evaluated as
The transition probability density of the BM under the condition that it stays forever in the positive region is then given by
for . The diffusion process whose transition probability density is given by (9) is called the 3-dimensional Bessel process, abbreviated as BES(3), by the following reasons. Consider the -dimensional BM, , whose coordinates are given by independent one-dimensional standard BMs . The distance from the origin of the -dimensional BM
can be regarded as a diffusion process in and its transition probability density is given by
where is the modified Bessel function with the gamma function . The process is called the -dimensional Bessel process, BES(), KS91 ; RY98 ; BS02 . The transition probability density (9) of the BM conditioned to stay positive is equal to (10) with , i.e., , since .
Consider the space of all configurations of particles in with a fixed order of positions,
which is called the Weyl chamber of type C in the representation theory FH91 . For , consider the following determinant
where the equality is given by the relation (9) and multilinearity of determinant. By the theory of Karlin and McGregor KM59 (see also KT05 with Lin73 and GV85 ), the probability that particle system of BES(3)’s starting from the configuration can keep the order of particle positions by avoiding any collision of particles for time period is given by
By the Markov property of diffusion processes, if we assume that the configuration at time is fixed to be , for , the transition probability density from at time to at time is given as
Let and define
for . Since we have known the asymptotics
for and , and
for and . The -particle system of noncolliding 3-dimensional Bessel bridges with duration 1 all starting from the origin is defined as the diffusion process such that its transition probability density is given by (13) and (14), and denoted by . That is, for any sequence of times , and for any sequence of regions ,
Note that no generality is lost by setting the time duration be 1, by the scaling property between space and time of the present -particle system inherited from BM via BES(3). From now on we call , simply the -noncolliding Bessel bridges for short. Remark that, if we set and in (14), and Eq.(1) is obtained.
ii.2 Matrix-valued Brownian bridge in symmetry class C
For , let be the unit matrix and define the matrix
where denotes the zero-matrix. Let and be the collections of all Hermitian matrices and all complex symmetric matrices, respectively. Then consider the following set of Hermitian matrices,
where denotes the transpose of and denotes the Hermitian conjugate of . We can see that any element satisfies the relation
which means that satisfies the symplectic Lie algebra; symbolically written as FH91 . Due to the additional symmetry (17), the real eigenvalues of are given in the form , where . Altland and Zirnbauer studied as the set of the Hamiltonians in the Bogoliubov-de Gennes formalism for the BCS mean-field theory of superconductivity, with regarding the pairing of positive and negative eigenvalues , as the particle-hole symmetry in the Bogoliubov-de Gennes theory AZ96 ; AZ97 . They called (a representation of) the symmetry class C, since is denoted by in Cartan’s notation Hel78 .
The Brownian bridge with duration 1 starting from the origin is defined as the one-dimensional standard BM starting from 0 conditioned to return to 0 at time and denoted by . The transition probability density of is given by
for . Let , and , be independent Brownian bridges with duration 1 starting from the origin. Put
for , and
and consider the matrices , and . Then the matrix-valued BM is defined by
In order to define , we have used independent Brownian bridges. By definition (19), , and . That is, can be regarded as a Brownian bridge in the -dimensional space .
At each time , can be diagonalized by a unitary-symplectic matrix and we can obtain the eigenvalue process with . Using the generalized Bru’s theorem given in KT04 , we can determine the transition probability density for the positive part of eigenvalue process . The result is exactly the same as Eqs.(13) and (14). It implies that , with probability one, and the present positive-eigenvalue process is equivalent in distribution with the noncolliding Bessel bridges . Figure 3 shows a sample of paths of the eigenvalue process , for generated by computer (see Sec.IV for detail). There we can see ten positive paths and their mirror images with respect to the line . The sample of paths of the noncolliding Bessel bridges, , shown in Fig.2 for is just obtained by the upper half of this figure.
For the -path system of noncolliding Bessel bridges, , we study the maximum values for each path attained in the time interval ,
The problem considered in the present paper is to clarify the statistical property of the random variables . We will report the exact expressions of the distribution function for the outermost path for general and the numerical results for inner paths. The dependence of is studied and the asymptotics in will be discussed.
Iii Exact Results for the Outermost Paths
iii.1 Distribution function of
In this section we derive an exact expression for the distribution function of the maximum value of the outermost path, . In order to that first we consider the absorbing BM in an interval for , in which two absorbing walls are put at the origin and at the position . The transition probability density for is the solution of the diffusion equation with the initial condition and with the Dirichlet boundary conditions By the method of separation of variables and the Fourier analysis, the unique solution is determined as (see, for example, KGV03 ),
for . Note that it is the different expression of the function
which was used in our previous paper KIK08 .
Consider the following two determinantal functions,
where . By the theory of Karlin and McGregor KM59 , (resp. ) is the probability for the -dimensional absorbing BM starting from (resp. ) at time to survive during time period by avoiding any hitting with the absorbing boundaries of (resp. ), and to arrive at (resp. ) at time . Note that if we think that the -dimensional vector represents the positions of particles on , a hitting with the boundary of means a hitting of the innermost particle with the origin or any collision between neighboring particles . Similarly a hitting with the boundary of means , or any collision of particles, or a hitting of the outer most particle with the wall at .
Consider the -particle system of noncolliding BES(3), starting from the configuration at time ; , and arriving at the configuration at time ; . Let . Then we can say that . By definition of the -noncolliding Bessel bridges, , given in Sec.II.A, we can conclude that
As shown in Appendix A, the method of the Schur function expansion KT04 gives the following asymptotics for and in ,
Lemma 1. For ,
Remark 1. This expression is exactly the same as Eq.(5) in SMCRF08 , which was derived by the path-integral method using a Selberg’s integral. Here we would like to put emphasis on the resemblance between the summand of (27) and (14). As mentioned in Sec.II.B, Eq.(14) is the same as the probability density of the eigenvalue-distribution of random matrices in the class C (with variance ). The exponent of the factor in (27) is the dimension of the space . Another evidence to show the hidden symmetry of the present maximum-value problem is the following. The character of the irreducible representation corresponding to a partition of the symplectic Lie algebra is given by FH91