相关论文: Non-D-finite excursions in the quarter plane

In part I (math.PR/0406392) we proved for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is of the maximal order square root of n. In higher dimensions we call…

We study the convex hull of the first $n$ steps of a planar random walk, and present large-$n$ asymptotic results on its perimeter length $L_n$, diameter $D_n$, and shape. In the case where the walk has a non-zero mean drift, we show that…

The fact that the euclidean algorithm eventually terminates is pervasive in mathematics. In the language of continued fractions, it can be stated by saying that the orbits of rational points under the Gauss map x-->{1/x} eventually reach…

In this note, we consider random walks in the quarter plane with arbitrary big jumps. We announce the extension to that class of models of the analytic approach of [G. Fayolle, R. Iasnogorodski, and V. Malyshev, Random walks in the quarter…

Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study the scaling limits of a random…

We consider a state-dependent, time-dependent, discrete random walks $X_t^{\{a_n\}}$ defined on natural numbers $\mathbb{N}$ (bent to a "stair" in $\mathbb{N}^2$) where the random walk depends on input of a positive deterministic sequence…

Deterministic walk in an excited random environment is a non-Markov integer-valued process $(X_n)_{n=0}^{\infty}$, whose jump at time $n$ depends on the number of visits to the site $X_n$. The environment can be understood as stacks of…

We consider a continuous-time random walk in the quarter plane for which the transition intensities are constant on each of the four faces $(0,\infty)^2$, $F_1=\{0\}\times(0,\infty)$, $F_2=(0,\infty)\times\{0\}$ and $\{(0,0)\}$. We show…

Given a finitely generated group with generating set $S$, we study the \emph{cogrowth} sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks…

Random flights (also called run-and-tumble walks or transport processes) represent finite velocity random motions changing direction at any Poissonian time. These models in d-dimension, can be studied giving a general formulation of the…

We study the area distribution of closed walks of length $n$, beginning and ending at the origin. The concept of area of a walk in the square lattice is generalized and the usefulness of the new concept is demonstrated through a simple…

We describe a new algebraic technique for enumerating self-avoiding walks on the rectangular lattice. The computational complexity of enumerating walks of $N$ steps is of order $3^{N/4}$ times a polynomial in $N$, and so the approach is…

Let $X_1, X_2, \ldots$ be a sequence of i.i.d. real-valued random variables with mean zero, and consider the scaled random walk of the form $Y^N_{k+1} = Y^N_{k} + a_N(Y^N_k) X_{k+1}$, where $a_N: \mathbb R \to \mathbb R_+$. We show, under…

We explore a continuous-time quantum walk starting at a single vertex on the discrete path and cycle with a cubic nonlinearity. Such nonlinearities arise in Bose-Einstein condensates described by the Gross-Pitaevskii equation or by…

Quantum walks (QWs) describe particles evolving coherently on a lattice. The internal degree of freedom corresponds to a Hilbert space, called coin system. We consider QWs on Cayley graphs of some group $G$. In the literature,…

We consider three kinds of discrete-time arrival processes: transient, intermediate and recurrent, characterized by a finite, possibly finite and infinite number of events, respectively. In this context, we study renewal processes which are…

On the trace of a discrete-time simple random walk on $\mathbb{Z}^d$ for $d\geq 2$, we consider the evolution of favorite sites, i.e., sites that achieve the maximal local time at a certain time. For $d=2$, we show that almost surely three…

The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighbouring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an…

The quantum walk (QW) is the term given to a family of algorithms governing the evolution of a discrete quantum system and as such has a founding role in the study of quantum computation. We contribute to the investigation of QW phenomena…

We study the random walk on the symmetric group $S_n$ generated by the conjugacy class of cycles of length $k$. We show that the convergence to uniform measure of this walk has a cut-off in total variation distance after $\frac{n}{k} log n$…