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

Katori, Makoto; Izumi, Minami; Kobayashi, Naoki

doi:10.1007/s10955-008-9524-0

Cited by 26 publications

(56 citation statements)

References 23 publications

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“…(5) yield back the result of Ref. [18]. For generic p, the probability distribution function (pdf) F ′ p (M ) is bellshaped, exhibiting a single mode.…”

Section: Introductionsupporting

confidence: 66%

“…For generic p, the probability distribution function (pdf) F ′ p (M ) is bellshaped, exhibiting a single mode. At variance with previous studies [18,19], our expression (5) is easily amenable to an asymptotic analysis for small M . Indeed, when M → 0, the leading contribution to the sum in (5) comes from n i = i and its p!…”

Section: Introductionmentioning

confidence: 87%

“…In particular, we are interested in the cumulative distribution [17,18,19,20,21] aiming at an analytical derivation of these estimates.…”

Section: Introductionmentioning

confidence: 99%

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Exact Distribution of the Maximal Height ofpVicious Walkers

et al. 2008

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“…(5) yield back the result of Ref. [18]. For generic p, the probability distribution function (pdf) F ′ p (M ) is bellshaped, exhibiting a single mode.…”

Section: Introductionsupporting

confidence: 66%

Section: Introductionmentioning

confidence: 87%

See 1 more Smart Citation

Exact Distribution of the Maximal Height ofpVicious Walkers

et al. 2008

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“…At t = t * it has continuous first and second order derivatives. By continuity the results of Theorem 2.4 and Corollary 2.5 continue to hold for a = 0, which is the case of non-intersecting squared Bessel bridges [35].…”

Section: Statement Of Resultsmentioning

confidence: 84%

“…[27,35,39,51] and below. In this paper we consider the case where all particles start at the same positive value a > 0 and end at 0.…”

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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Critical behavior of nonintersecting Brownian motions at a tacnode

Delvaux

Kuijlaars

Zhang

2011

Comm Pure Appl Math

129

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We study a model of n one-dimensional non-intersecting Brownian motions with two prescribed starting points at time t = 0 and two prescribed ending points at time t = 1 in a critical regime where the paths fill two tangent ellipses in the time-space plane as n → ∞. The limiting mean density for the positions of the Brownian paths at the time of tangency consists of two touching semicircles, possibly of different sizes. We show that in an appropriate double scaling limit, there is a new familiy of limiting determinantal point processes with integrable correlation kernels that are expressed in terms of a new RiemannHilbert problem of size 4 × 4. We prove solvability of the Riemann-Hilbert problem and establish a remarkable connection with the Hastings-McLeod solution of the Painlevé II equation. We show that this Painlevé II transcendent also appears in the critical limits of the recurrence coefficients of the multiple Hermite polynomials that are associated with the non-intersecting Brownian motions. Universality suggests that the new limiting kernels apply to more general situations whenever a limiting mean density vanishes according to two touching square roots, which represents a new universality class.

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

Cited by 26 publications

References 23 publications

Exact Distribution of the Maximal Height ofpVicious Walkers

Exact Distribution of the Maximal Height ofpVicious Walkers

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

Critical behavior of nonintersecting Brownian motions at a tacnode

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