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 Szegő-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 noncrossing 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.
Funding Statement
The first author was supported by EPSRC as part of the MASDOC DTC, Grant. No. EP/HO23364/1.
Acknowledgments
We are grateful to Thomas Bothner for many useful discussions. Some results of the present paper were reported at the conference “Randomness and Symmetry” held at University College Dublin in June 18–22, 2018 (see also the thesis [17]).
Citation
Will FitzGerald. Roger Tribe. Oleg Zaboronski. "Asymptotic expansions for a class of Fredholm Pfaffians and interacting particle systems." Ann. Probab. 50 (6) 2409 - 2474, November 2022. https://doi.org/10.1214/22-AOP1586
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