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Title: Asymptotics of noncolliding q-exchangeable random walks
Abstract We consider a process of noncollidingq-exchangeable random walks on Z making steps 0 (‘straight’) and −1 (‘down’). A single random walk is calledq-exchangeable if under an elementary transposition of the neighboring steps ( down , straight ) ( straight , down ) the probability of the trajectory is multiplied by a parameter q ( 0 , 1 ) . Our process ofmnoncollidingq-exchangeable random walks is obtained from the independentq-exchangeable walks via the Doob’sh-transform for a nonnegative eigenfunctionh(expressed via theq-Vandermonde product) with the eigenvalue less than 1. The system ofmwalks evolves in the presence of an absorbing wall at 0. The repulsion mechanism is theq-analogue of the Coulomb repulsion of random matrix eigenvalues undergoing Dyson Brownian motion. However, in our model, the particles are confined to the positive half-line and do not spread as Brownian motions or simple random walks. We show that the trajectory of the noncollidingq-exchangeable walks started from an arbitrary initial configuration forms a determinantal point process, and express its kernel in a double contour integral form. This kernel is obtained as a limit from the correlation kernel ofq-distributed random lozenge tilings of sawtooth polygons. In the limit as m , q = e γ / m withγ > 0 fixed, and under a suitable scaling of the initial data, we obtain a limit shape of our noncolliding walks and also show that their local statistics are governed by the incomplete beta kernel. The latter is a distinguished translation invariant ergodic extension of the two-dimensional discrete sine kernel.  more » « less
Award ID(s):
1664617 2153869
PAR ID:
10443922
Author(s) / Creator(s):
;
Publisher / Repository:
IOP Publishing
Date Published:
Journal Name:
Journal of Physics A: Mathematical and Theoretical
Volume:
56
Issue:
36
ISSN:
1751-8113
Page Range / eLocation ID:
Article No. 365203
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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