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Register a new userExamples of interacting particle systems on $mathbb{Z}$ as Pfaffian point processes: annihilating and coalescing random walks
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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.
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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 that of Activated Random Walks. Long-range effects intrinsic to the conservative dynamics and lack of a simple algebraic structure cause standard tools and techniques to break down. This makes the mathematical study of this model remarkably challenging. Yet, some exciting progress has been made in the last ten years, with the development of a framework of tools and methods which is finally becoming more structured. In these lecture notes we present the existing results and reproduce the techniques developed so far.
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 restricted to the infinite percolation cluster ${{mathcal{C}}_{infty}}$ of a supercritical Bernoulli percolation. When no more than $k$ individuals per site are allowed, we obtain the $k$-type contact process, which can be derived from the branching random walk by killing all particles that are born at a site where already $k$ individuals are present. We prove that local survival of the branching random walk on ${{mathbb{Z}}^d}$ also implies that for $k$ sufficiently large the associated $k$-type contact process survives on ${{mathcal{C}}_{infty}}$. This implies that the strong critical parameters of the branching random walk on ${{mathbb{Z}}^d}$ and on ${{mathcal{C}}_{infty}}$ coincide and that their common value is the limit of the sequence of strong critical parameters of the associated $k$-type contact processes. These results are extended to a family of restrained branching random walks, that is, branching random walks where the success of the reproduction trials decreases with the size of the population in the target site.
Our first result states that the orthogonal and symplectic Bessel processes are rigid in the sense of Ghosh and Peres. Our argument in the Bessel case proceeds by an estimate of the variance of additive statistics in the spirit of Ghosh and Peres. Second, a sufficient condition for number rigidity of stationary Pfaffian processes, relying on the Kolmogorov criterion for interpolation of stationary processes and applicable, in particular, to pfaffian sine-processes, is given in terms of the asymptotics of the spectral measure for additive statistics.
We study Markov chains on $mathbb Z^m$, $mgeq 2$, that behave like a standard symmetric random walk outside of the hyperplane (membrane) $H={0}times mathbb Z^{m-1}$. The transition probabilities on the membrane $H$ are periodic and also depend on the incoming direction to $H$, what makes the membrane $H$ two-sided. Moreover, sliding along the membrane is allowed. We show that the natural scaling limit of such Markov chains is a $m$-dimensional diffusion whose first coordinate is a skew Brownian motion and the other $m-1$ coordinates is a Brownian motion with a singular drift controlled by the local time of the first coordinate at $0$. In the proof we utilize a martingale characterization of the Walsh Brownian motion and determine the effective permeability and slide direction. Eventually, a similar convergence theorem is established for the one-sided membrane without slides and random iid transition probabilities.
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 version of the first-named authors ``Meeting time lemma~ that bounds the probability of random walk hitting a deterministic trajectory in terms of hitting times of static vertices. The meeting time result is then used to bound the expected full coalescence time of multiple random walks over a graph. This last theorem is a discrete-time version of a result by the first-named author, which had been previously conjectured by Aldous and Fill. Our bounds improve on recent results by Lyons and Oveis-Gharan; Kanade et al; and (in certain regimes) Cooper et al.
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