Random Walks with Invariant Loop Probabilities: Stereographic Random Walks

Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. After a general introduction, we focus our attention on the elliptic case: random walks on a circle with built-in reflexing boundaries.     

Open Quantum Random Walks

A new model of quantum random walks is introduced, on lattices as well as on finite graphs. These quantum random walks take into account the behavior of open quantum systems. They are the exact quantum analogues of classical Markov chains. We explore the “quantum trajectory” point of view on these quantum random walks, that is, we show that measuring the position of the particle after each time-step gives rise to a classical Markov chain, on the lattice times the state space of the particle. This quantum trajectory is a simulation of the master equation of the quantum random walk. The physical pertinence of such quantum random walks and the way they can be concretely realized is discussed. Differences and connections with the already well-known quantum random walks, such as the Hadamard random walk, are established.     

Random Time-Dependent Quantum Walks

We consider the discrete time unitary dynamics given by a quantum walk on the lattice \mathbb Z^d{\mathbb {Z}^d} performed by a quantum particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in \mathbb Z^d{\mathbb {Z}^d} when the sequence of unitary updates is given by an i.i.d. sequence of random matrices. When averaged over the randomness, this distribution is shown to display a drift proportional to the time and its centered counterpart is shown to display a diffusive behavior with a diffusion matrix we compute. A moderate deviation principle is also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. A generalization to unitary updates distributed according to a Markov process is also provided.     

Persistent Random Walks in Stationary Environment

We study the behavior of persistent random walks (RW) on the integers in a random environment. A complete characterization of the almost sure limit behavior of these processes, including the law of large numbers, is obtained. This is done in a general situation where the environmental sequence of random variables is stationary and ergodic. Szász and Tóth obtained a central limit theorem when the ratio /, of right- and left-transpassing probabilities satisfies /a<1 a.s. (for a given constant a). We consider the case where / has wider fluctuations; we shall observe that an unusual situation arises: the RW may converge a.s. to infinity even with zero drift. Then, we obtain nonclassical limiting distributions for the RW. Proofs are based on the introduction of suitable branching processes in order to count the steps performed by the RW.     

Lingering Random Walks in Random Environment on a Strip

We consider a recurrent random walk (RW) in random environment (RE) on a strip. We prove that if the RE is i. i. d. and its distribution is not supported by an algebraic subsurface in the space of parameters defining the RE then the RW exhibits the (log t)² asymptotic behaviour. The exceptional algebraic subsurface is described by an explicit system of algebraic equations. One-dimensional walks with bounded jumps in a RE are treated as a particular case of the strip model. If the one dimensional RE is i. i. d., then our approach leads to a complete and constructive classification of possible types of asymptotic behaviour of recurrent random walks. Namely, the RW exhibits the (log t)² asymptotic behaviour if the distribution of the RE is not supported by a hyperplane in the space of parameters which shall be explicitly described. And if the support of the RE belongs to this hyperplane then the corresponding RW is a martingale and its asymptotic behaviour is governed by the Central Limit Theorem.     

Microscopic Approach to Random Walks

In the present paper the microscopic approach to random walk models is introduced. For any particular model it provides a rigorous way to derive the transport equations for the macroscopic density of walking particles. Although it is not more complicated than the standard random walk framework it has virtually no limitations with respect to the initial distribution of particles. As a consequence, the transport equations derived with this method almost automatically give answers to such important problems as aging and two point probability distribution.     

Loop-Erased Walks and Random Matrices

Journal of Statistical Physics - It is well known that there are close connections between non-intersecting processes in one dimension and random matrices, based on the reflection principle. There...     

Quantum Random Walks and Thermalisation

It is shown how to construct quantum random walks with particles in an arbitrary faithful normal state. A convergence theorem is obtained for such walks, which demonstrates a thermalisation effect: the limit cocycle obeys a quantum stochastic differential equation without gauge terms. Examples are presented which generalise that of Attal and Joye (J Funct Anal 247:253–288, 2007).     

Simple Random Walks on Tori

We consider a Markov chain whose phase space is a d-dimensional torus. A point x jumps to x+ with probability p(x) and to x– with probability 1–p(x). For Diophantine and smooth p we prove that this Markov chain has an absolutely continuous invariant measure and the distribution of any point after n steps converges to this measure.     

Equidistribution of Random Walks on Spheres

We study the equidistribution on spheres of the n-step transition probabilities of random walks on graphs. We give sufficient conditions for this property being satisfied and for the weaker property of asymptotical equidistribution. We analyze the asymptotical behaviour of the Green function of the simple random walk on ² and we provide a class of random walks on Cayley graphs of groups, whose transition probabilities are not even asymptotically equidistributed.     

Random Walks in Space Time Mixing Environments

We prove that random walks in random environments, that are exponentially mixing in space and time, are almost surely diffusive, in the sense that their scaling limit is given by the Wiener measure.     

Random Walks on a Fractal Solid

It is established that the trapping of a random walker undergoing unbiased, nearest-neighbor displacements on a triangular lattice of Euclidean dimension d=2 is more efficient (i.e., the mean walklength n before trapping of the random walker is shorter) than on a fractal set, the Sierpinski tower, which has a Hausdorff dimension D exactly equal to the Euclidean dimension of the regular lattice. We also explore whether the self similarity in the geometrical structure of the Sierpinski lattice translates into a self similarity in diffusional flows, and find that expressions for n having a common analytic form can be obtained for sites that are the first- and second-nearest-neighbors to a vertex trap.     

Area Distribution for Directed Random Walks

We study the probability distribution for the area under a directed random walk in the plane. The walk can serve as a simple model for avalanches based on the idea that the front of an avalanche can be described by a random walk and the size is given by the area enclosed. This model captures some of the qualitative features of earthquakes, avalanches, and other self-organized critical phenomena in one dimension. By finding nonlinear functional relations for the generating functions we calculate directly the exponent in the size distribution law and find it to be 4/3.     

Continuous-Time Random Walks under Finite Concentrations

Journal of Experimental and Theoretical Physics - Nonlinear equations generalizing the continuous-time random walk model to the case of finite concentrations have been derived. The equations take...     

Classification on the Average of Random Walks

During the last decade many attempts have been made to characterize absence of spontaneous breaking of continuous symmetry for the Heisenberg model on graphs by using suitable classifications of random walks (refs. 4 and 10). We propose and study a new type problem for random walks on graphs, which is particularly interesting for disordered graphs. We compare this classification with the classical one and with an analogous one introduced in ref. 4. Various examples, that are not space-homogeneous, are provided.     

Dynamical Localization of Quantum Walks in Random Environments

We study shock statistics in the scalar conservation law ∂ _t u+∂ _x f(u)=0, x∈ℝ, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process u(x,t), x∈ℝ. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u ²/2). This suggests the kinetic equations of shock clustering are completely integrable.     

Locally Perturbed Random Walks with Unbounded Jumps

Szász and Telcs (J. Stat. Phys. 26(3), 1981) have shown that for the diffusively scaled, simple symmetric random walk, weak convergence to the Brownian motion holds even in the case of local impurities if d≥2. The extension of their result to finite range random walks is straightforward. Here, however, we are interested in the situation when the random walk has unbounded range. Concretely we generalize the statement of Szász and Telcs (J. Stat. Phys. 26(3), 1981) to unbounded random walks whose jump distribution belongs to the domain of attraction of the normal law. We do this first: for diffusively scaled random walks on Z ^d (d≥2) having finite variance; and second: for random walks with distribution belonging to the non-normal domain of attraction of the normal law. This result can be applied to random walks with tail behavior analogous to that of the infinite horizon Lorentz-process; these, in particular, have infinite variance, and convergence to Brownian motion holds with the superdiffusive \(\sqrt{n\log n}\) scaling.