Studying Self-Organized Criticality with Exactly Solved Models

This is a somewhat expanded version of the notes of a series of lectures given at Lausanne and Stellenbosch in 1998-99. They are intended to provide a pedagogical introduction to the abelian sandpile model of self-organized criticality, and its related models : the q=0 state Potts model, Takayasu aggregation model, the voter model, spanning trees, Eulerian walkers model etc. It provides an overview of the known results, and explains the equivalence of these models. Some open questions are discussed in the concluding section.Comment: Latex with epsf, 47 pages, 14 figure

Logarithmic two-point correlators in the Abelian sandpile model

We present the detailed calculations of the asymptotics of two-site correlation functions for height variables in the two-dimensional Abelian sandpile model. By using combinatorial methods for the enumeration of spanning trees, we extend the well-known result for the correlation $\sigma_{1,1} \simeq 1/r^4$ of minimal heights $h_1=h_2=1$ to $\sigma_{1,h} = P_{1,h}-P_1P_h$ for height values $h=2,3,4$. These results confirm the dominant logarithmic behaviour $\sigma_{1,h} \simeq (c_h\log r + d_h)/r^4 + {\cal O}(r^{-5})$ for large $r$, predicted by logarithmic conformal field theory based on field identifications obtained previously. We obtain, from our lattice calculations, the explicit values for the coefficients $c_h$ and $d_h$ (the latter are new).Comment: 28 page

Probability around the Quantum Gravity. Part 1: Pure Planar Gravity

In this paper we study stochastic dynamics which leaves quantum gravity equilibrium distribution invariant. We start theoretical study of this dynamics (earlier it was only used for Monte-Carlo simulation). Main new results concern the existence and properties of local correlation functions in the thermodynamic limit. The study of dynamics constitutes a third part of the series of papers where more general class of processes were studied (but it is self-contained), those processes have some universal significance in probability and they cover most concrete processes, also they have many examples in computer science and biology. At the same time the paper can serve an introduction to quantum gravity for a probabilist: we give a rigorous exposition of quantum gravity in the planar pure gravity case. Mostly we use combinatorial techniques, instead of more popular in physics random matrix models, the central point is the famous $\alpha =-7/2$ exponent.Comment: 40 pages, 11 figure

Phase Transitions on Nonamenable Graphs

We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees

Conditions for recurrence and transience for one family of random walks

In the present paper, a family of two-dimensional random walks $\{\mathbf{S}_t,\mathcal{A}\}$ in the main quarter plane, where $\mathcal{A}$ is a set of infinite sequences of real values, is studied. For $a\in\mathcal{A}$, a random walk is denoted $\mathbf{S}_t(a)$ $=(S_t^{(1)}(a),$ $S_t^{(2)}(a))$. Let $\theta$ denote the infinite sequence of zeros. For $a\neq\theta$ the components $S_t^{(1)}(a)$ and $S_t^{(2)}(a)$ are assumed to be correlated in the specified way that is defined exactly in the paper, while for $a=\theta$, the random walk $\mathbf{S}_t(\theta)$ is the simple two-dimensional random walk in the main quarter plane. We derive the conditions on $a$ under which a random walk $\mathbf{S}_t(a)$ is recurrent or transient. In addition, we introduce new classes of random walks, $\psi$-random walks, and derive conditions under which a subfamily of random walks $\{\mathbf{S}_t,\mathcal{A}_\psi\}$, $\mathcal{A}_\psi\subset\mathcal{A}$ belongs to the class of $\psi$-random walks.Comment: Substantially revised paper, 35 pages, 10pt, 2 figure

Gibbs and Quantum Discrete Spaces

Gibbs measure is one of the central objects of the modern probability, mathematical statistical physics and euclidean quantum field theory. Here we define and study its natural generalization for the case when the space, where the random field is defined is itself random. Moreover, this randomness is not given apriori and independently of the configuration, but rather they depend on each other, and both are given by Gibbs procedure; We call the resulting object a Gibbs family because it parametrizes Gibbs fields on different graphs in the support of the distribution. We study also quantum (KMS) analog of Gibbs families. Various applications to discrete quantum gravity are given.Comment: 37 pages, 2 figure