TY - JOUR

T1 - Asymptotic expansions for a class of Fredholm Pfaffians and interacting particle systems

AU - Fitzgerald, Will

AU - Tribe, Roger

AU - Zaboronski, Oleg

PY - 2022/8/2

Y1 - 2022/8/2

N2 - Motivated by the phenomenon of duality for interacting particle systems we introduce two classes of Pfaffian kernels describing a number of Pfaffian point processes in the `bulk' and at the `edge'. Using the probabilistic method due to Mark Kac, we prove two Szeg\H{o}-type asymptotic expansion theorems for the corresponding Fredholm Pfaffians. The idea of the proof is to introduce an effective random walk with transition density determined by the Pfaffian kernel, express the logarithm of the Fredholm Pfaffian through expectations with respect to the random walk, and analyse the expectations using general results on random walks. We demonstrate the utility of the theorems by calculating asymptotics for the empty interval and non-crossing probabilities for a number of examples of Pfaffian point processes: coalescing/annihilating Brownian motions, massive coalescing Brownian motions, real zeros of Gaussian power series and Kac polynomials, and real eigenvalues for the real Ginibre ensemble.

AB - Motivated by the phenomenon of duality for interacting particle systems we introduce two classes of Pfaffian kernels describing a number of Pfaffian point processes in the `bulk' and at the `edge'. Using the probabilistic method due to Mark Kac, we prove two Szeg\H{o}-type asymptotic expansion theorems for the corresponding Fredholm Pfaffians. The idea of the proof is to introduce an effective random walk with transition density determined by the Pfaffian kernel, express the logarithm of the Fredholm Pfaffian through expectations with respect to the random walk, and analyse the expectations using general results on random walks. We demonstrate the utility of the theorems by calculating asymptotics for the empty interval and non-crossing probabilities for a number of examples of Pfaffian point processes: coalescing/annihilating Brownian motions, massive coalescing Brownian motions, real zeros of Gaussian power series and Kac polynomials, and real eigenvalues for the real Ginibre ensemble.

UR - https://projecteuclid.org/journals/annals-of-probability/volume-50/issue-6/Asymptotic-expansions-for-a-class-of-Fredholm-Pfaffians-and-interacting/10.1214/22-AOP1586.short

U2 - 10.48550/arXiv.2107.14504

DO - 10.48550/arXiv.2107.14504

M3 - Article

SN - 0091-1798

JO - Annals of Probability

JF - Annals of Probability

ER -