Examples of interacting particle systems on $mathbb{Z}$ as Pfaffian point processes: annihilating and coalescing random walks

A class of interacting particle systems on $mathbb{Z}$, involving instantaneously annihilating or coalescing nearest neighbour random walks, are shown to be Pfaffan point processes for all deterministic initial conditions. As diffusion limits, explicit Pfaffan kernels are derived for a variety of coalescing and annihilating Brownian systems. For Brownian motions on $mathbb{R}$, depending on the initial conditions, the corresponding kernels are closely related to the bulk and edge scaling limits of the Pfaffan point process for real eigenvalues for the real Ginibre ensemble of random matrices. For Brownian motions on $mathbb{R}_{+}$ with absorbing or reflected boundary conditions at zero new interesting Pfaffan kernels appear. We illustrate the utility of the Pfaffan structure by determining the extreme statistics of the rightmost particle for the purely annihilating Brownian motions, and also computing the probability of overcrowded regions for all models.

What is the probability that a large random matrix has no real eigenvalues?

We study the large-$n$ limit of the probability $p_{2n,2k}$ that a random $2ntimes 2n$ matrix sampled from the real Ginibre ensemble has $2k$ real eigenvalues. We prove that, $$lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,2k}=lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,0}= -frac{1}{sqrt{2pi}}zetaleft(frac{3}{2}right),$$ where $zeta$ is the Riemann zeta-function. Moreover, for any sequence of non-negative integers $(k_n)_{ngeq 1}$, $$lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,2k_n}=-frac{1}{sqrt{2pi}}zetaleft(frac{3}{2}right),$$ provided $lim_{nrightarrow infty} left(n^{-1/2}log(n)right) k_{n}=0$.