A Pearson Random Walk with Steps of Uniform Orientation and Dirichlet Distributed Lengths

A constrained diffusive random walk of n steps in ℝ ^d and a random flight in ℝ ^d , which are equivalent, were investigated independently in recent papers (J. Stat. Phys. 127:813, 2007; J. Theor. Probab. 20:769, 2007, and J. Stat. Phys. 131:1039, 2008). The n steps of the walk are independent and identically distributed random vectors of exponential length and uniform orientation. Conditioned on the sum of their lengths being equal to a given value l, closed-form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d=1,2,4. Uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2D and of two steps in 4D.     

Twice Scattered Particles in a Plane Are Uniformly Distributed

It was recently shown (Physica A 216:299–315, 1995) that in two dimensions the sum of three vectors each of whose lengths is exponentially distributed, whose direction is uniformly distributed and such that the sum of their lengths is l, is uniformly distributed on a disk of radius l. We state here this random walk result in terms of scattering of particles as follows: in two dimensions twice isotropically scattered particles by random (i.e., Poisson distributed) scatterers are uniformly distributed. We show that there is no other dimension d and no other number of scatterings s for which the corresponding result (i.e., uniform distribution on a d-dimensional sphere after s scatterings) holds.     

A New Family of Solvable Pearson-Dirichlet Random Walks

An n-step Pearson-Gamma random walk in ? ^d starts at the origin and consists of n independent steps with gamma distributed lengths and uniform orientations. The gamma distribution of each step length has a shape parameter q>0. Constrained random walks of n steps in ? ^d are obtained from the latter walks by imposing that the sum of the step lengths is equal to a fixed value. Simple closed-form expressions were obtained in particular for the distribution of the endpoint of such constrained walks for any d≥d ₀ and any n≥2 when q is either \(q = \frac{d}{2} - 1 \) (d ₀=3) or q=d?1 (d ₀=2) (Le Caër in J. Stat. Phys. 140:728–751, 2010). When the total walk length is chosen, without loss of generality, to be equal to 1, then the constrained step lengths have a Dirichlet distribution whose parameters are all equal to q and the associated walk is thus named a Pearson-Dirichlet random walk. The density of the endpoint position of a n-step planar walk of this type (n≥2), with q=d=2, was shown recently to be a weighted mixture of 1+floor(n/2) endpoint densities of planar Pearson-Dirichlet walks with q=1 (Beghin and Orsingher in Stochastics 82:201–229, 2010). The previous result is generalized to any walk space dimension and any number of steps n≥2 when the parameter of the Pearson-Dirichlet random walk is q=d>1. We rely on the connection between an unconstrained random walk and a constrained one, which have both the same n and the same q=d, to obtain a closed-form expression of the endpoint density. The latter is a weighted mixture of 1+floor(n/2) densities with simple forms, equivalently expressed as a product of a power and a Gauss hypergeometric function. The weights are products of factors which depends both on d and n and Bessel numbers independent of d.     

Random walk on a one-dimensional lattice with a random distribution of traps

The probability of return to the starting point of a particle executing a random walk on a one-dimensional lattice with a static distribution of traps, is derived, for asymptotically large n, the number of steps. The probability decreases as exp ($\text{?n}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$), in agreement with some recent estimates of its upper and lower bounds for diffusion in d dimensions.     

The non-isotropic two-dimensional random walk

Abstract

A formula is obtained for the joint probability density function of the angle and length of the resultant of an N-step non-isotropic random walk in two dimensions for arbitrary step angle and radius probability density and for any fixed number of steps. The problem is attacked by applying the theory of generalized functions concentrated on smooth manifolds. The analysis is presented initially for the case where only the angles are random. The characteristic function is defined for the walk in terms of angular and radial frequencies and the inversion is obtained in terms of a sum of Hankel transforms. The Hankel transform sum is transformed by showing that it can be interpreted in terms of the motions of the two-dimensional Euclidean plane corresponding to the rotations and translations resulting from a sequence of fixed steps. This transformation results in an expression involving integrations over two manifolds defined by delta functions. The properties of the manifolds defined by the delta functions are then considered and this results in some simplification of the formulae. The analysis is then generalized to the case where both the phase and length of each step in the walk are random. Finally, seven examples are presented including the general two-step walk and three walks which lead to generalized K density functions for the resultant.     

Random walks on a lattice

This paper discusses the mean-square displacement for a random walk on a two-dimensional lattice, whose transitions to nearest-neighbor sites are symmetric in the horizontal and vertical directions and depend on the column currently occupied. Under a uniform density condition for the step probabilities it is shown that the horizontal mean-square displacement aftern steps is asymptotically proportional ton, and independent of the particular column configuration. The model generalizes that of Seshadri, Lindenberg, and Shuler and the arguments are essentially probabilistic.     

Random Walk on the Range of Random Walk

We study the random walk X on the range of a simple random walk on ℤ ^d in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin.     

Variance of the range of a random walk

T _n, the expectation of the square of the number of distinct sites occupied by a random walk in steps 1 throughn, is obtained from its relation to the dual first occupancy probabilityF _ij(x, x), and the latter quantity is obtained from a recursion with the first occupancy probabilityF _k (x). The varianceV _n of the number of distinct sites occupied is calculated directly from T_n; the procedure is illustrated by the calculation ofV _n (4096 /n) and the derivation of asymptotic expansions forV _n for a particular random walk in dimensions 1 through 3.Work completed under the auspices of the United States Department of Energy.     

The fractal nature of molecular trajectories in fluids

The scaled lengths of molecular trajectories obtained by molecular dynamics simulation of a hard-sphere fluid are shown to have the same fractal dimensionD=2 as the random walk. Self-similarity first appears on length scales typically a factor of 25 greater than the mean free-path length, whereas for the simple random walk with constant step size the onset occurs after only six steps; the reason for the slow convergence is shown to be the near exponential distribution of intercollision path lengths of the fluid molecules. The influence of density on the scaled path lengths is also discussed.     

Spectral Dimension and Random Walks on the Two Dimensional Uniform Spanning Tree

We study the simple random walk on the uniform spanning tree on \mathbb Z²{\mathbb {Z}^2} . We obtain estimates for the transition probabilities of the random walk, the distance of the walk from its starting point after n steps, and exit times of both Euclidean balls and balls in the intrinsic graph metric. In particular, we prove that the spectral dimension of the uniform spanning tree on \mathbb Z²{\mathbb {Z}^2} is 16/13 almost surely.     

Transience, Recurrence and Critical Behavior¶for Long-Range Percolation

We study the behavior of the random walk on the infinite cluster of independent long-range percolation in dimensions d= 1,2, where x and y are connected with probability . We show that if d<s<2d, then the walk is transient, and if s≥ 2d, then the walk is recurrent. The proof of transience is based on a renormalization argument. As a corollary of this renormalization argument, we get that for every dimension d≥ 1, if d>s>2d, then there is no infinite cluster at criticality. This result is extended to the free random cluster model. A second corollary is that when d≥& 2 and d>s>2d we can erase all long enough bonds and still have an infinite cluster. The proof of recurrence in two dimensions is based on general stability results for recurrence in random electrical networks. In particular, we show that i.i.d. conductances on a recurrent graph of bounded degree yield a recurrent electrical network. Received: 27 October 2000 / Accepted: 29 November 2001     

The Birth of the Infinite Cluster:¶Finite-Size Scaling in Percolation

We address the question of finite-size scaling in percolation by studying bond percolation in a finite box of side length n, both in two and in higher dimensions. In dimension d= 2, we obtain a complete characterization of finite-size scaling. In dimensions d>2, we establish the same results under a set of hypotheses related to so-called scaling and hyperscaling postulates which are widely believed to hold up to d= 6. As a function of the size of the box, we determine the scaling window in which the system behaves critically. We characterize criticality in terms of the scaling of the sizes of the largest clusters in the box: incipient infinite clusters which give rise to the infinite cluster. Within the scaling window, we show that the size of the largest cluster behaves like n ^d π _n , where π _n is the probability at criticality that the origin is connected to the boundary of a box of radius n. We also show that, inside the window, there are typically many clusters of scale n ^d π _n , and hence that “the” incipient infinite cluster is not unique. Below the window, we show that the size of the largest cluster scales like ξ ^d π_ξ log(n/ξ), where ξ is the correlation length, and again, there are many clusters of this scale. Above the window, we show that the size of the largest cluster scales like n ^d P _∞, where P _∞ is the infinite cluster density, and that there is only one cluster of this scale. Our results are finite-dimensional analogues of results on the dominant component of the Erdős–Rényi mean-field random graph model. Received: 6 December 2000 / Accepted: 25 May 2001     

On ergodic properties for harmonic crystals

We consider the dynamics of a harmonic crystal in n dimensions with d components, where d and n are arbitrary, d, n ⩾ 1. The initial data are given by a random function with finite mean energy density which also satisfies a Rosenblatt-or Ibragimov-type mixing condition. The random function is close to diverse space-homogeneous processes as x _n → ±∞, with the distributions μ±. We prove that the phase flow is mixing with respect to the limit measure of statistical solutions. Partially supported by RFBR under grant no. 06-01-00096.     

Mott Law as Lower Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic stationary simple point process on ℝ^d, d≥2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.     

Scaling of Loop-Erased Walks in 2 to 4 Dimensions

We simulate loop-erased random walks on simple (hyper-)cubic lattices of dimensions 2, 3 and 4. These simulations were mainly motivated to test recent two loop renormalization group predictions for logarithmic corrections in d=4, simulations in lower dimensions were done for completeness and in order to test the algorithm. In d=2, we verify with high precision the prediction D=5/4, where the number of steps n after erasure scales with the number N of steps before erasure as n∼N ^D/2. In d=3 we again find a power law, but with an exponent different from the one found in the most precise previous simulations: D=1.6236±0.0004. Finally, we see clear deviations from the naive scaling n∼N in d=4. While they agree only qualitatively with the leading logarithmic corrections predicted by several authors, their agreement with the two-loop prediction is nearly perfect.     

Trapping in the Random Conductance Model

We consider random walks on ? ^d among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but whose support extends all the way to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point at time 2n. We show that in the situations when the heat kernel exhibits subdiffusive decay—which is known to occur in dimensions d≥4—the walk gets trapped for a time of order n in a small spatial region. This shows that the strategy used earlier to infer subdiffusive lower bounds on the heat kernel in specific examples is in fact dominant. In addition, we settle a conjecture concerning the worst possible subdiffusive decay in four dimensions.     

Localization for random potentials supported on a subspace

We consider the lattice Schrödinger operator acting onl ² ( ^d ) with random potential (independent, identically distributed random variables), supported on a subspace of dimension 1 v <d. We use the multiscale analyses scheme to prove that this operator exhibits exponential localization at the edges of the spectrum for any disorder or outside the interval [-2d, 2d] for sufficiently high disorder.     

Mott Law as Upper Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic simple point process on , d ≥ 2, furnished with independent energy marks. The jump rates of the random walk decay exponentially in the jump length and depend on the energy marks via a Boltzmann–type factor. This is an effective model for the phonon–induced hopping of electrons in disordered solids in the regime of strong Anderson localization. Under some technical assumption on the point process we prove an upper bound for the diffusion matrix of the random walk in agreement with Mott law. A lower bound for d ≥ 2 in agreement with Mott law was proved in [8].     

Random walk in a random multiplicative environment

We investigate the dynamics of a random walk in a random multiplicative medium. This results in a random, but correlated, multiplicative process for the spatial distribution of random walkers. We show how the details of these correlations determine the asymptotic properties of the walk, i.e., the central limit theorem does not apply to these multiplicative processes. We also study a periodic source-trap medium in which a unit cell contains one source, followed byL–1 traps. We calculate the asymptotic behavior of the number of particles, and determine the conditions for which there is growth or decay in this average number. Finally, we discuss the asymptotic behavior of a random walk in the presence of randomly distributed, partially-absoprbing traps. For this case, a temporal regime of purely exponential decay of the density can occur, before the asymptotic stretched exponential decay, exp(–at ^1/3), sets in.     

Diffusion of a Heteropolymer in a Multi-Interface Medium

We consider a heteropolymer, consisting of an i.i.d. concatenation of hydrophilic and hydrophobic monomers, in the presence of water and oil arranged in alternating layers. The heteropolymer is modelled by a directed path ( $\left( {i,S_i } \right)_{i \in \mathbb{N}_0 }$ , where the vertical component lives on $\mathbb{Z}$ , and the layers are horizontal with equal width. The path measure for the vertical component is given by that of simple random walk multiplied by an exponential weight factor that favors matches and disfavors mismatches between the monomers and the medium. We study the vertical motion of the heteropolymer as a function of its total length n when the width of the layers is d _n and the parameters in the exponential weight factor are such that the heteropolymer tends to stay close to an interface (“localized regime”). In the limit as n→∞ and under the condition that lim _n→∞ d _n /log log n=∞ and lim _n→∞ d _n /log n=0, we show that the vertical motion is a diffusive hopping between neighboring interfaces on a time scale exp[χd _n (1+o(1))], where χ is computed explicitly in terms of a variational problem. An analysis of this variational problem sheds light on the optimal hopping strategy.