Non-colliding Brownian bridges and the asymmetric tacnode process

Abstract

We consider non-colliding Brownian bridges starting from two points and returning to the same position. These positions are chosen such that, in the limit of large number of bridges, the two families of bridges just touch each other forming a tacnode. We obtain the limiting process at the tacnode, the "asymmetric tacnode process". It is a determinantal point process with correlation kernel given by two parameters: (1) the curvature's ratio λ>0 of the limit shapes of the two families of bridges, (2) a parameter σcontrolling the interaction on the fluctuation scale. This generalizes the result for the symmetric tacnode process (λ=1 case).

21 pages, 1 figure, LaTeX; Includes a further representation of the kernel

References in corpus (6)

• The Pearcey Process
• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
• Fluctuations of a one-dimensional polynuclear growth model in a half space
• Nonintersecting random walks in the neighborhood of a symmetric tacnode
• Non-colliding Brownian Motions and the extended tacnode process

Cited by in corpus (19)

• Nonintersecting random walks in the neighborhood of a symmetric tacnode
• Nonintersecting Brownian motions on the unit circle
• Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
• Multiplicative functionals on ensembles of non-intersecting paths
• A periodic hexagon tiling model and non-Hermitian orthogonal polynomials
• The hard edge tacnode process and the hard edge Pearcey process with non-intersecting squared Bessel paths
• Non-intersecting squared Bessel paths at a hard-edge tacnode
• The gap probabilities of the tacnode, Pearcey and Airy point processes, their mutual relationship and evaluation
• The hard-edge tacnode process for Brownian motion
• Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume
• Asymptotics of the Tacnode process: a transition between the gap probabilities from the Tacnode to the Airy process
• Transitions between critical kernels: from the tacnode kernel and critical kernel in the two-matrix model to the Pearcey kernel
• Fluctuations of the Arctic curve in the tilings of the Aztec diamond on restricted domains
• Coupled GUE-minor Processes
• Edge Universality for Nonintersecting Brownian Bridges
• The tacnode Riemann-Hilbert problem
• Tacnode GUE-minor Processes and Double Aztec Diamonds
• Two Lax systems for the Painlevé II equation, and two related kernels in random matrix theory
• Nonintersecting Brownian bridges between reflecting or absorbing walls