Nearest-neighbor Random Walk Research Articles

Einstein relation for random walks in random environments

We consider a tracer particle performing a nearest neighbor random walk on Z d in dimension d ⩾ 3 with random jump rates. This kind of a walk models the motion of a charged particle under a constant external electric field. We assume that the jump rates admit only two values 0 < γ - < γ + < + ∞ , representing the lower and upper conductivities. We prove the existence of the mobility coefficient and that it equals to the diffusivity coefficient of the particle in zero external field.

Random Walks on Groups and Monoids with a Markovian Harmonic Measure

We consider a transient nearest neighbor random walk on a group $G$ with finite set of generators $S$. The pair $(G,S)$ is assumed to admit a natural notion of normal form words where only the last letter is modified by multiplication by a generator. The basic examples are the free products of a finitely generated free group and a finite family of finite groups, with natural generators. We prove that the harmonic measure is Markovian of a particular type. The transition matrix is entirely determined by the initial distribution which is itself the unique solution of a finite set of polynomial equations of degree two. This enables to efficiently compute the drift, the entropy, the probability of ever hitting an element, and the minimal positive harmonic functions of the walk. The results extend to monoids.

Multiple Shocks in Bricklayers' Model

In bricklayers' model, which is a generalization of the misanthrope processes, we show that a nontrivial class of product distributions is closed under the time-evolution of the process. This class also includes measures fitting to shock data of the limiting PDE. In particular, we show that shocks of this type with discontinuity of size one perform ordinary nearest neighbor random walks only interacting, in an attractive way, via their jump rates. Our results are related to those of Belitsky and Schutz(4) on the simple exclusion process, although we do not use quantum formalism as they do. The structures we find are described from a fixed position. Similar ones were found in Balazs,(2) valak as seen from the random position of the second class particle.

The homotopical reduction of a nearest neighbor random walk*

Consider a nearest neighbor random walk on a graph G and discard all the segments of its trajectory that are homotopically equivalent to a single point. We prove that if the lift of the random walk to the covering tree of G is transient, then the resulting “reduced” trajectories induce a Markov chain on the set of oriented edges of G. We study this chain in relation with the original random walk. As an intermediate result, we give a simple proof of the Markovian structure of the harmonic measure on trees.

On the surviving probability of an annihilating branching process and application to a nonlinear voter model

We study a discrete time interacting particle system which can be considered as an annihilating branching process on Z where at each time every particle either performs a jump as a nearest neighbor random walk, or splits (with probability ε) into two particles which will occupy the nearest neighbor sites. Furthermore, if two particles come to the same site, then they are removed from the system. We show that if the branching probability ε>0 is small enough, and the number of particles at initial time is finite, then the surviving probability, i.e. the probability p( t) that there is at least one particle at time t decays to zero exponentially fast. This result is applied to a nonlinear discrete time voter model (in a random and nonrandom environment) obtained as a small perturbation with parameter ε of the classical voter model. For this class of models, we show that if ε>0 is small enough, then the process converges to a unique invariant probability measure independently on the initial distribution. It is known that the classical one-dimensional voter model (in a random environment as well as without environment) is not ergodic, that is there exist at least two extremal invariant probability measures. Our results prove therefore the phase transition in ε=0.

Statistics of the depth probed by cw measurements of photons in a turbid medium

Photon migration in a turbid medium has been modeled in many different ways. The motivation for such modeling is based on technology that can be used to probe potentially diagnostic optical properties of biological tissue. Surprisingly, one of the more effective models is also one of the simplest. It is based on statistical properties of a nearest-neighbor lattice random walk. Here we develop a theory allowing one to calculate the number of visits by a photon to a given depth, if it is eventually detected at an absorbing surface. This mimics cw measurements made on biological tissue and is directed towards characterizing the depth reached by photons injected at the surface. Our development of the theory uses formalism based on the theory of a continuous-time random walk (CTRW). Formally exact results are given in the Fourier-Laplace domain, which, in turn, are used to generate approximations for parameters of physical interest.

Numerical Simulations of Random Walk in Random Environment

This paper is concerned with the numerical simulation of a random walk in a random environment in dimension d = 2. Consider a nearest neighbor random walk on the 2-dimensional integer lattice. The transition probabilities at each site are assumed to be themselves random variables, but fixed for all time. This is the random environment. Consider a parallel strip of radius R centered on an axis through the origin. Let XR be the probability that the walk that started at the origin exits the strip through one of the boundary lines. Then XR is a random variable, depending on the environment. In dimension d = 1, the variable XR converges in distribution to the Bernoulli variable, X∞ = 0, 1 with equal probability, as R → ∞. Here the 2-dimensional problem is studied using Gauss-Seidel and multigrid algorithms.

Random Walks On Finite Convex Sets Of Lattice Points

This paper examines the convergence of nearest-neighbor random walks on convex subsets of the latticesℤd. The main result shows that for fixedd, O(γ2) steps are sufficient for a walk to “get random,” where γ is the diameter of the set. Toward this end a new definition of convexity is introduced for subsets of lattices, which has many important properties of the concept of convexity in Euclidean spaces.

The Cut-Off Phenomenon for Random Walks on Hamming Graphs with Variable Growth Conditions

The cut-off phenomenon is a remarkable critical phenomenon observed in the process of convergence to equilibrium in a wide variety of Markov chains. Diaconis–Graham–Morrison [3] established the first precise evaluation around the critical time for Ehrenfests' urn model concerning 2 urns and d balls, i.e. nearest neighbor random walks on hypercube \mathbf Z^d . They showed that deviation from the equilibrium state is described well by using the error functiƒon. In this article, we work out the evaluation around the critical time for simple random walks on Hamming graphs H(d,n) , which coincide with an extended Ehrenfests' urn model concerning n urns and d balls. In our case, not only d but also n can grow in several manners. If n/d tends to 0 , the similar result to [3] remains valid and microscopic deviation from the equilibrium state is described by the error function. If n/d tends to a nonzero constant, however, it is shown that the error function has to be replaced by an expression involving Poisson distributions.

On self-attracting $d$-dimensional random walks

Let $\{X_t\}_{t \geq 0}$ be a symmetric, nearest-neighbor random walk on $\mathbb{Z}^d$ with exponential holding times of expectation $1/d$, starting at the origin. For a potential $V: \mathbb{Z}^d \to [0, \infty)$ with finite and nonempty support, define transformed path measures by $d \hat{\mathbb{P}}_T \equiv \exp (T^{-1} \int_0^T \int_0^T V(X_s - X_t) ds dt) d \mathbb{P}/Z_T$ for $T > 0$, where $Z_T$ is the normalizing constant. If $d = 1$ or if the self-attraction is sufficiently strong, then $||X_t||_{\infty}$ has an exponential moment under $\hat{\mathbb{P}}_T$ which is uniformly bounded for $T > 0$ and $t \in [0, T]$. We also prove that ${X_t}_{t \geq 0}$ under suitable subsequences of $\{\hat{\mathbb{P}}_T\}_{T > 0}$ behaves for large $T$ asymptotically like a mixture of space-inhomogeneous ergodic random walks. For special cases like a sufficiently strong Dirac-type interaction, we even prove convergence of the transformed path measures and the law of $X_T$ as well as of the law of the empirical measure $L_T$ under $\{\hat{\mathbb{P}}_T\}_{T > 0}$.

On the norms of the random walks on planar graphs

We consider the nearest neighbor random walk on planar graphs. For certain families of these graphs, we give explicit upper bounds on the norm of the random walk operator in terms of the minimal number of edges at each vertex. We show that for a wide range of planar graphs the spectral radius of the random walk is less than one.

Optimal running times for systems of random walks involving several particles

In this paper we consider a system of several particles together with a set of stopping criteria. Each unit of time we are allowed to move exactly one particle which then performs a nearest neighbor random walk. We analyze strategies which minimize or maximize the expected running times until one of the stopping criteria is fulfilled

A remark on diffusion of directed polymers in random environments

We consider a system of random walks or directed polymers interacting with an environment which is random in space and time. Under minimal assumptions on the distribution of the environment, we prove that this system has diffusive behavior with probability one if d > 2 and fl < flo, where flo is defined in terms of the probability that the symmetric nearest neighbor random walk on the d-dimensional integer lattice ever returns to its starting point. We also obtain a precise estimate for the mean square displacement of this system.

Random walks on random simple graphs

This paper looks at random regular simple graphs and considers nearest neighbor random walks on such graphs. This paper considers walks where the degree d of each vertex is around (log n)a where a is a constant which is at least 2 and where n is the number of vertices. By extending techniques of Dou, this paper shows that for most such graphs, the position of the random walk becomes close to uniformly distributed after slightly more than log n/log d steps. This paper also gets similar results for the random graph G(n, p), where p = d/(n − 1). © 1996 John Wiley & Sons, Inc.

The "True" Self-Avoiding Walk with Bond Repulsion on $\mathbb{Z}$: Limit Theorems

The true self-avoiding walk with bond repulsion is a nearest neighbor random walk on $\mathbb{Z}$, for which the probability of jumping along a bond of the lattice is proportional to $\exp(-g \cdot$ number of previous jumps along that bond). First we prove a limit theorem for the distribution of the local time process of this walk. Using this result, later we prove a local limit theorem, as $A \rightarrow \infty$, for the distribution of $A^{-2/3}X_{\theta_{s/A}}$, where $\theta_{s/A}$ is a random time distributed geometrically with mean $e^{-s/A}(1 - e^{-s/A})^{-1} = A/s + O(1)$. As a by-product we also obtain an apparently new identity related to Brownian excursions and Bessel bridges.

The nearest neighbor random walk on subspaces of a vector space and rate of convergence

LetX be the collection ofk-dimensional subspaces of ann-dimensional vector spaceVn overGF(q). A metric may be defined onX by letting $$d\left( {W_k ,V_k } \right) = k - \dim \left( {W_k \cap V_k } \right)forW_k ,V_k \in X$$ The nearest neighbor random walk onX starts at an initial fixed pointZk ofX and, from wherever it finds itself, moves with the same probability to any of the points ofX at distance one from it. We analyze the rate of convergence of the nearest neighbor random walk onX to its stationary distribution. The argument involves lifting the process onX to a random walk onGLn(GF(q)) and using the Fourier transform andq-Hahn polynomials.

Spectra of graphs and fractal dimensions II

This paper continues the study of exponentsd(x), d ω (x), d R (x) andd μ (x) for graphG; and the nearest neighbor random walk {X n } n∈N onG, if the starting pointX 0=x is fixed. These exponents are responsible for the geometric, resistance, diffusion and spectral properties of the graph. The main concern of this paper is the relation of these exponents to the spectral density of the transition matrix. A series of new exponentse, e ω ,e R ,e μ are introduced by allowingx to vary along the vertices. The results suggest that the geometric and resistance properties of the graph are responsible for the diffusion speed on the graph.

‘True’ self-avoiding walks with generalized bond repulsion on ℤ

We consider a nearest-neighbor random walk on ℤ, for which the probability of jumping along a bond of the lattice is proportional to exp[−g. (number of previous jumps along that bond) k ], withg>0,k∈(0,1]. After a review of earlier results obtained for the casek=1 we outline the generalizations fork∈(0,1), obtaining a whole range of anomalous diffusion limits.

On three conjectures by K. E. Shuler

Some fifteen years ago, Shuler formulated three conjectures relating to the large-time asymptotic properties of a nearest-neighbor random walk on ℤ2 that is allowed to make horizontal steps everywhere but vertical steps only on a random fraction of the columns. We give a proof of his conjectures for the situation where the column distribution is stationary and satisfies a certain mixing codition. We also prove a strong form of scaling to anisotropic Brownian motion as well as a local limit theorem. The main ingredient of the proofs is a large-deviation estimate for the number of visits to a random set made by a simple random walk on ℤ. We briefly discuss extensions to higher dimension and to other types of random walk.

Reaction efficiency of diffusion-controlled processes on finite, planar arrays.

In this paper we investigate the reaction efficiency of diffusion-controlled processes on finite, planar arrays having physical or chemical receptors. This problem translates into the statistical-mechanical one of examining the geometrical factors affecting the trapping of a random walker on small lattices of dimension d=2, having N sites and average valency \ensuremath{\nu}\ifmmode\bar\else\textasciimacron\fi{}. Extensive calculations of the site-specific average walk length 〈n〉 before trapping, a measure of the efficiency of the underlying diffusion-reaction process, have been carried out on triangular, square-planar, hexagonal, and Penrose platelets for N=16 and N=48. From the variety of distinct lattices considered, and the data generated, three general conclusions can be drawn. First, for fixed N, the smaller the number ${\mathit{N}}_{\mathit{b}}$ of vertices defining the boundary of the finite lattice under consideration, the smaller the value of the (overall) average walk length 〈n\ifmmode\bar\else\textasciimacron\fi{}〉 of the random walker before trapping. Second, for fixed N and fixed ${\mathit{N}}_{\mathit{b}}$, the smaller the value of the (overall) root-mean-square distance (r${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{2}$${)}^{1/2}$ of the N lattice sites relative to the center of the array, the smaller the value of 〈n\ifmmode\bar\else\textasciimacron\fi{}〉. Third, for fixed {N,${\mathit{N}}_{\mathit{b}}$,(r${\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}^{2}$${)}^{1/2}$}, 〈n\ifmmode\bar\else\textasciimacron\fi{}〉 decreases with an increase in the (overall) average valency \ensuremath{\nu}\ifmmode\bar\else\textasciimacron\fi{} of lattice sites comprising the array. Thus there are similarities but also real and significant differences in the conclusions drawn here in studying stochastic processes taking place on small, finite lattices of arbitrary shape and those found in studying nearest-neighbor random walks on infinite, periodic lattices of unit cells characterized by a given (N,d,\ensuremath{\nu}). We comment on these and on the possible relevance of this work to one aspect of morphogenesis, viz., predicting the morphologies assumed by small platelets when growth is optimized with respect to (chemical or physical) signal processing at receptor sites.