Directed Paths in Random Media

The aim of this article is to give a pedagogical introduction to the exact equilibrium and nonequilibrium properties of free fermionic quantum spin chains. In a first part we present in full details the canonical diagonalisation procedure and review quickly the equilibrium dynamical properties. The phase diagram is analysed and possible phase transitions are discussed. The two next chapters are concerned with the effect of aperiodicity and quenched disorder on the critical pr...

We report an implementation of the recursion method that addresses quantum many-body dynamics in the nonperturbative regime. The implementation has two key ingredients: a computer-algebraic routine for symbolic calculation of nested commutators and a procedure to extrapolate the sequence of Lanczos coefficients according to the universal operator growth hypothesis. We apply the method to calculate infinite-temperature correlation functions for spin-$1/2$ systems on one- and t...

We consider the behavior of an Ising ferromagnet obeying the Glauber dynamics under the influence of a fast switching, random external field. In Part I, we introduced a general formalism for describing such systems and presented the mean field theory. In this article we derive results for the one dimensional case, which can be only partially solved. Monte Carlo simulations performed on a square lattice indicate that the main features of the mean field theory survive the prese...

The study by Glauber of the time-dependent statistics of the Ising chain is extended to the case where each spin is influenced unequally by its nearest neighbours. The asymmetry of the dynamics implies the failure of the detailed balance condition. The functional form of the rate at which an individual spin changes its state is constrained by the global balance condition with respect to the equilibrium measure of the Ising chain. The local magnetization, the equal-time and tw...

Sinai's model of diffusion in one-dimension with random local bias is studied by a real space renormalization group which yields asymptotically exact long time results. The distribution of the position of a particle and the probability of it not returning to the origin are obtained, as well as the two-time distribution which exhibits "aging" with $\frac{\ln t}{\ln t'}$ scaling and a singularity at $\ln t =\ln t'$. The effects of a small uniform force are also studied. Extensi...

Random walks on general graphs play an important role in the understanding of the general theory of stochastic processes. Beyond their fundamental interest in probability theory, they arise also as simple models of physical systems. A brief survey of the physical relevance of the notion of random walk on both undirected and directed graphs is given followed by the exposition of some recent results on random walks on randomly oriented lattices. It is worth noticing that gene...

This article aims to provide an introductory survey on quantum random walks. Starting from a physical effect to illustrate the main ideas we will introduce quantum random walks, review some of their properties and outline their striking differences to classical walks. We will touch upon both physical effects and computer science applications, introducing some of the main concepts and language of present day quantum information science in this context. We will mention recent d...

We demonstrate the universality of the spectral correlation functions of a QCD inspired random matrix model that consists of a random part having the chiral structure of the QCD Dirac operator and a deterministic part which describes a schematic temperature dependence. We calculate the correlation functions analytically using the technique of Itzykson-Zuber integrals for arbitrary complex super-matrices. An alternative exact calculation for arbitrary matrix size is given for ...

We consider the Ising model and the directed walk on two-dimensional layered lattices and show that the two problems are inherently related: The zero-field thermodynamical properties of the Ising model are contained in the spectrum of the transfer matrix of the directed walk. The critical properties of the two models are connected to the scaling behavior of the eigenvalue spectrum of the transfer matrix which is studied exactly through renormalization for different self-simil...

We study the problem of a random walk on a lattice in which bonds connecting nearest neighbor sites open and close randomly in time, a situation often encountered in fluctuating media. We present a simple renormalization group technique to solve for the effective diffusive behavior at long times. For one-dimensional lattices we obtain better quantitative agreement with simulation data than earlier effective medium results. Our technique works in principle in any dimension, al...