• Corpus ID: 236635367

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

@inproceedings{FitzGerald2021AsymptoticEF,
  title={Asymptotic expansions for a class of Fredholm Pfaffians and interacting particle systems},
  author={Will FitzGerald and Roger Tribe and Oleg V. Zaboronski},
  year={2021}
}
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… 

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References

SHOWING 1-10 OF 42 REFERENCES
Sharp asymptotics for Fredholm Pfaffians related to interacting particle systems and random matrices
It has been known since the pioneering paper of Mark Kac, that the asymptotics of Fredholm determinants can be studied using probabilistic methods. We demonstrate the efficacy of Kac' approach by
Pfaffian Formulae for One Dimensional Coalescing and Annihilating Systems
The paper considers instantly coalescing, or instantly annihilating, systems of one-dimensional Brownian particles on the real line. Under maximal entrance laws, the distribution of the particles at
Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble
It is known that the bulk scaling limit of the real eigenvalues for the real Ginibre ensemble is equal in distribution to the rescaled t → ∞ ?> limit of the annihilation process A + A → ∅ ?> .
Examples of Interacting Particle Systems on $$\mathbb {Z}$$Z as Pfaffian Point Processes: Annihilating and Coalescing Random Walks
A class of interacting particle systems on $$\mathbb {Z}$$Z, involving instantaneously annihilating or coalescing nearest neighbour random walks, are shown to be Pfaffian point processes for all
Universal Painlevé VI Probability Distribution in Pfaffian Persistence and Gaussian First-Passage Problems with a sech-Kernel
We recast the persistence probability for the spin located at the origin of a half-space arbitrarily $m$-magnetized Glauber-Ising chain as a Fredholm Pfaffian gap probability generating function with
Examples of Interacting Particle Systems on $$\mathbb {Z}$$ as Pfaffian Point Processes: Coalescing–Branching Random Walks and Annihilating Random Walks with Immigration
Two classes of interacting particle systems on $$\mathbb {Z}$$Z are shown to be Pfaffian point processes, at any fixed time and for all deterministic initial conditions. The first comprises
The true self-repelling motion
Abstract. We construct and study a continuous real-valued random process, which is of a new type: It is self-interacting (self-repelling) but only in a local sense: it only feels the self-repellance
From Painlevé to Zakharov–Shabat and beyond: Fredholm determinants and integro-differential hierarchies
As Fredholm determinants are more and more frequent in the context of stochastic integrability, we unveil the existence of a common framework in many integrable systems where they appear. This
Edge Distribution of Thinned Real Eigenvalues in the Real Ginibre Ensemble
This paper is concerned with the explicit computation of the limiting distribution function of the largest real eigenvalue in the real Ginibre ensemble when each real eigenvalue has been removed
Exact Persistence Exponent for the 2D-Diffusion Equation and Related Kac Polynomials.
TLDR
It is shown that the probability q_{0}(n) that Kac's polynomials, of (even) degree n, have no real root decays, for large n, as q¬0 (n)∼n^{-3/4}.
...
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