Related papers: Polya's random walk theorem

This paper presents the strong law of large numbers for a function of the local times of a transient random walk on groups, extending the research of Asymont and Korshunov for random walks on the integer lattice $\mathbb{Z}^d$. Under some…

We study a model of multi-excited random walk on a regular tree which generalizes the models of the once excited random walk and the digging random walk introduced by Volkov (2003). We show the existence of a phase transition of the…

We derive a quenched invariance principle for random walks in random environments whose transition probabilities are defined in terms of weighted cycles of bounded length. To this end, we adapt the proof for random walks among random…

There is a condition (T'), such that it is the necessary condition that a random walk in random environment is ballistic. Under this condition, we show the law of the iterated logarithm for a random walk in random environment.

In this paper we define direct product of graphs and give a recipe for obtained probability of observing particle on vertices in the continuous-time classical and quantum random walk. In the recipe, the probability of observing particle on…

Working in combinatorial model $\mathrm{W_{co}}(d)$, $d=1,2,\dots$, of P\'olya's random walker in $\mathbb{Z}^d$, we prove two theorems on recurrence to a vertex. We obtain an effective version of the first theorem if $d=2$. Using a…

We give a simple proof of the Fourier Inversion Theorem, using the methods of nonstandard analysis.

We prove a quenched functional central limit theorem for a one-dimensional random walk driven by a simple symmetric exclusion process. This model can be viewed as a special case of the random walk in a balanced random environment, for which…

Random transvections generate a walk on the space of symplectic forms on $\mathbf{F}_q^{2n}$. The main result is establishing cutoff for this Markov chain. After $n+c$ steps, the walk is close to uniform while before $n-c$, it is far from…

In [3] the radius of convergence of the generating function of the collision local time of two independent copies of an irreducible, symmetric and transient random walk on Zd, d \geq 1, was studied. Two versions were considered: z1, the…

The purpose of this paper is to prove directly, by an elementary method, the Poisson probability law. This proof is offered as an alternative to the more usual derivation from binomial distribution in the limit of small probabilities. The…

Let $S_n$ be a centered random walk with a finite variance, and define the new sequence $A_n:=\sum_{i=1}^n S_i$, which we call an integrated random walk. We are interested in the asymptotics of $$p_N:=P(\min_{1 \le k \le N} A_k \ge 0)$$ as…

The paper is devoted to an invariance principle for Kemperman's model of oscillating random walk on $\mathbb{Z}$. This result appears as an extension of the invariance principal theorem for classical random walks on $\mathbb{Z}$ or…

A new proof is given for the formula for the expected return time of a random walk on a graph. This proof makes use of known relationships between electric resistance and random walks.

In this note, we discuss a general definition of quantum random walks on graphs and illustrate with a simple graph the possibility of very different behavior between a classical random walk and its quantum analogue. In this graph,…

Let $S_n =X_1+\cdots +X_n$ be an irreducible random walk (r.w.) on the one dimensional integer lattice with zero mean, infinite variance and i.i.d. increments $X_n$. We obtain an upper and lower bounds of the potential function, $a(x)$, of…

We give an alternative proof of the localization of Sinai's random walk in random environment under weaker hypothesis than the ones used by Sinai. Moreover we give estimates that are stronger than the one of Sinai on the localization…

We study random walk on complex networks with transition probabilities which depend on the current and previously visited nodes. By using an absorbing Markov chain we derive an exact expression for the mean first passage time between pairs…

We give three derivations of Polya's approximation for the expected range of a simple random walk in one dimension. This result allows for an estimation of the volatility of a financial instrument from the difference between the high and…

Classical and quantum walks on some finite paths are introduced. It is shown that these walks have explicit solutions given in terms of exceptional Krawtchouk polynomials and their properties are explored. In particular, fractional revival…