Noncolliding Brownian Motion and Determinantal Processes

Katori, Makoto; Tanemura, Hideki

doi:10.1007/s10955-007-9421-y

Cited by 86 publications

(158 citation statements)

References 98 publications

(170 reference statements)

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“…, is equivalent with the BM conditioned to stay positive, and this process is realized as an h-transform of the absorbing BM with a wall at the origin (see, for example, [19]), we will see that…”

Section: Reflection Principle and Karlin-mcgregor Formulamentioning

confidence: 95%

“…Noncolliding diffusion particle systems are interesting and important statistical-mechanical processes, since they are related to the group representation-theory, the random matrix theory, and the exactly solved nonequilibrium statistical-mechanical models (e.g., ASEP and polynuclear growth models) [19]. The present system of noncolliding Bessel bridges is related to the class C ensemble of random matrices discussed by Altland and Zirnbauer [1,2] (see Sect.V.C of [17]) and it is a special case with parameters (ν, κ) = (1/2, 3) of the noncolliding generalized meanders [18] (see also [24]).…”

Section: X(t) ≡ |B(t)|mentioning

confidence: 97%

“…As demonstrated in [19], the noncolliding diffusion processes can be regarded as the multivariate extensions of Bessel processes. The present study suggests the possibility that the connection between the conditional BMs and the number theoretical functions (e.g., the Jacobi theta function, the Riemann zeta functions, and Dirichlet series) reported in [25,4] will be extended to many particle and multivariate systems.…”

Section: X(t) ≡ |B(t)|mentioning

confidence: 99%

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Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function

2008

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It is known that the moments of the maximum value of a one-dimensional conditional Brownian motion, the three-dimensional Bessel bridge with duration 1 started from the origin, are expressed using the Riemann zeta function. We consider a system of two Bessel bridges, in which noncolliding condition is imposed. We show that the moments of the maximum value is then expressed using the double Dirichlet series, or using the integrals of products of the Jacobi theta functions and its derivatives. Since the present system will be provided as a diffusion scaling limit of a version of vicious walker model, the ensemble of 2-watermelons with a wall, the dominant terms in longtime asymptotics of moments of height of 2-watermelons are completely determined. For the height of 2-watermelons with a wall, the average value was recently studied by Fulmek by a method of enumerative combinatorics.

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Section: Reflection Principle and Karlin-mcgregor Formulamentioning

confidence: 95%

Section: X(t) ≡ |B(t)|mentioning

confidence: 97%

Section: X(t) ≡ |B(t)|mentioning

confidence: 99%

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Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function

2008

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“…Deift [22], Johansson [31], Katori and Tanemura [37], Borodin and Olshanski [11], and many other papers cited therein. See also the surveys of Soshnikov [49], König [38], Hough et al [30], and Johansson [32].…”

Section: Introductionmentioning

confidence: 99%

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

Kuijlaars

Martínez–Finkelshtein

Wielonsky

2008

Commun. Math. Phys.

120

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We study a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = T at x = 0. In the limit n → ∞, after appropriate rescaling, the paths fill out a region in the tx-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at x = 0, but at a certain critical time t * the smallest paths hit the hard edge and from then on are stuck to it. For t = t * we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a 3 × 3 matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large n limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.

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“…It can be generalized to spacetime systems and if all spatio-temporal correlation functions are given by determinants, the process is also said to be determinantal [9,15]. The noncolliding Brownian motion with a finite number of particles N is determinantal for all deterministic initial configurations ξ(·) = N j=1 δ r j (·).…”

Section: Introductionmentioning

confidence: 99%

System of Complex Brownian Motions Associated with the O’Connell Process

Katori

2012

J Stat Phys

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The O'Connell process is a softened version (a geometric lifting with a parameter a > 0) of the noncolliding Brownian motion such that neighboring particles can change the order of positions in one dimension within the characteristic length a. This process is not determinantal. Under a special entrance law, however, Borodin and Corwin gave a Fredholm determinant expression for the expectation of an observable, which is a softening of an indicator of a particle position. We rewrite their integral kernel to a form similar to the correlation kernels of determinantal processes and show, if the number of particles is N , the rank of the matrix of the Fredholm determinant is N . Then we give a representation for the quantity by using an N -particle system of complex Brownian motions (CBMs). The complex function, which gives the determinantal expression to the weight of CBM paths, is not entire, but in the combinatorial limit a → 0 it becomes an entire function providing conformal martingales and the CBM representation for the noncolliding Brownian motion is recovered.

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Noncolliding Brownian Motion and Determinantal Processes

Cited by 86 publications

References 98 publications

Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function

Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function

Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

System of Complex Brownian Motions Associated with the O’Connell Process

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