Oleg Zaboronski (Warwick University)
Asymptotic expansions for Fredholm pfaffians and interacting particle
systems
Abstract:
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 Szego-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. This is a joint work with Roger Tribe and Will
FitzGerald.