Sharp asymptotics for Fredholm Pfaffians related to interacting particle systems and random matrices

It has been known since the pioneering paper of Mark Kac [20], that the asymptotics of Fredholm determinants can be studied using probabilistic methods. We demonstrate the efficacy of Kac’ approach by studying the Fredholm Pfaffian describing the statistics of both non-Hermitian random matrices and annihilating Brownian motions. Namely, we establish the following two results. Firstly, let √N+λ ₘₐₓ be the largest real eigenvalue of a random N× N matrix with independent $N(0,1)$ entries (the ‘real Ginibre matrix’). Consider the limiting N→ ∞ distribution P[λ ₘₐₓ<-L] of the shifted maximal real eigenvalue λ ₘₐₓ. Then \[ lim L→ ∞ e^{ {1}{2√2π }ζ (3/2 )L} P (λ ₘₐₓ<-L ) =e^{Cₑ}, \] where ζ is the Riemann zeta-function and \[ Cₑ=1/2log 2+1/4π ∑ ₙ₌₁∞1/n (-π +∑ ₘ₌₁ⁿ⁻¹ {1}{√m(n-m)} ). \] Secondly, let Xₜ⁽ᵐᵃˣ⁾ be the position of the rightmost particle at time t for a system of annihilating Brownian motions (ABM’s) started from every point of R₋. Then \[ lim L→ ∞ e^{ {1}{2√2π }ζ (3/2 )L} P ( {Xₜ⁽ᵐᵃˣ⁾}{√4t}<-L ) =e^{Cₑ}. \] These statements are a sharp counterpart of the results of [22], improved by computing the O(L⁰) term in the asymptotic L→ ∞ expansion of the corresponding Fredholm Pfaffian.