Related papers: Heat kernel estimates for strongly recurrent rando…
A convergence theorem is obtained for quantum random walks with particles in an arbitrary normal state. This result unifies and extends previous work on repeated-interactions models, including that of the author (2010, J. London Math. Soc.…
We study the existence and uniqueness of the heat kernel on infinite, locally finite, connected graphs. For general graphs, a uniqueness criterion, shown to be optimal, is given in terms of the maximal valence on spheres about a fixed…
This paper studies anchored expansion, a non-uniform version of the strong isoperimetric inequality. We show that every graph with i-anchored expansion contains a subgraph with isoperimetric (Cheeger) constant at least i. We prove a…
We will investigate a random mass splitting model and the closely related random walk in a random environment (RWRE). The heat kernel for the RWRE at time t is the mass splitting distribution at t. We prove a quenched invariance principle…
We discuss a particular kind of quantum walk on a general graph. We affix two semi-infinite lines to a general finite graph, which we call tails. On the tails, the particle making the walk simply advances one unit at each time step, so that…
This paper concerns the long-term behaviour of a system of interacting random walks labeled by vertices of a finite graph. The model is reversible which allows to use the method of electric networks in the study. In addition, examples of…
For $d\geq 2$, we establish the existence and uniqueness of heat kernels for a large class of time-dependent second order diffusion operator with jumps, which is the sum of time-dependent of a second order elliptic differential operators…
We show that anomalous diffusion can result when the steps of a random walk are not statistically independent. We present an algorithm that counts all the possible paths of particles diffusing on random graphs with arbitrary degree…
In this article, we consider the problem of estimating the heat kernel on measure-metric spaces equipped with a resistance form. Such spaces admit a corresponding resistance metric that reflects the conductivity properties of the set. In…
We compute the heat kernel coefficients that are needed for the regularization and renormalization of massive gravity. Starting from the Stueckelberg action for massive gravity, we determine the propagators of the different fields (massive…
The well known bunkbed conjecture about percolation on finite graphs is now resolved; Gladkov, Pak and Zimin, building upon work of Hollom, have constructed a counterexample. We revisit this conjecture and study it in the broader context of…
Let $(G,\mu)$ be a uniformly elliptic random conductance graph on $\mathbb{Z}^d$ with a Poisson point process of particles at time $t=0$ that perform independent simple random walks. We show that inside a cube $Q_K$ of side length $K$, if…
A close relation between hitting times of the simple random walk on a graph, the Kirchhoff index, resistance-centrality, and related invariants of unicyclic graphs is displayed. Combining with the graph transformations and some other…
We apply results from Baryshnikov, Brady, Bressler and Pemantle (2008) to compute limiting probability profiles for various quantum random walks in one and two dimensions. Using analytic machinery we show some features of the limit…
Simple random walks are a basic staple of the foundation of probability theory and form the building block of many useful and complex stochastic processes. In this paper we study a natural generalization of the random walk to a process in…
In the study of small and large networks it is customary to perform a simple random walk, where the random walker jumps from one node to one of its neighbours with uniform probability. The properties of this random walk are intimately…
For any recurrent random walk (S_n)_{n>0} on R, there are increasing sequences (g_n)_{n>0} converging to infinity for which (g_n S_n)_{n>0} has at least one finite accumulation point. For one class of random walks, we give a criterion on…
Maximization of the entropy rate is an important issue to design diffusion processes aiming at a well-mixed state. We demonstrate that it is possible to construct maximal-entropy random walks with only local information on the graph…
In recent years, protocols that are based on the properties of random walks on graphs have found many applications in communication and information networks, such as wireless networks, peer-to-peer networks and the Web. For wireless…
In this paper, we first give a direct proof for two recurrence relations of the heat kernels for hyperbolic spaces in \cite{DM}. Then, by similar computation, we give two similar recurrence relations of the heat kernels for spheres.…