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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.
Some stochastic systems are particularly interesting as they exhibit critical behavior without fine-tuning of a parameter, a phenomenon called self-organized criticality. In the context of driven-dissipative steady states, one of the main models is t
In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on ${{mathbb{Z}}^d}$ survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restr
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We prove new results on lazy random walks on finite graphs. To start, we obtain new estimates on return probabilities $P^t(x,x)$ and the maximum expected hitting time $t_{rm hit}$, both in terms of the relaxation time. We also prove a discrete-time v