Random Grammars

Projet MEVALThis is the first part in a series of papers, where we consider new connections between computer science and modern mathematical physics. Here we begin to study a class of "concrete" random processes covering most of well known processes, such as locally interacting processes, random fractals, random walks, queueing networks, random Turing machines, etc. Here we restrict ourselves to linear graphs. We establish existence and uniqueness of the dynamics in the thermodynamic limit and prove that this dynamics is clustering. We get ergodicity and non-recurrence conditions in a small perturbation region. We study invariant measures and large time fractal type behaviour for random context free grammars and languages

Interacting Strings of Characters

Projet MEVALWe consider several interacting strings of characters. Otherwise speaking, we consider one-dimensional random walks interacting with each other and with their environments (mostly in a one-sided manner). This embraces many applications: queueing networks with different customer types, random walks on some discrete non-commutative groups, random Turing machines and others. We present a scheme for the general theory of such processes, the general theory of random walks in the orthant being a particular case. This scheme is being developed in several papers, a review of which is presented here

Networks and dynamical systems

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Stabilization laws for process with a localised interaction

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Microscopic Models for Chemical Thermodynamics

We introduce an infinite particle system dynamics, which includes stochastic chemical kinetics models, the classical Kac model and free space movement. We study energy redistribution between two energy types (kinetic and chemical) in different time scales, similar to energy redistribution in the living organisms. One example is considered in great detail, where the model provides main formulas of chemical thermodynamics

Quantum Grammars. Part IV.1: KMS States on Quantum Grammars

Projet MEVALThis is the fourth part of the series of papers devoted to some relationships between the computer science and the quantum gravity. We consider quantum (unitary) continuous time evolution of spins on a lattice together with quantum evolution of the lattice itself. In physics such evolution was discussed in connection with the quantum gravity. It is also related to what is called quantum circuits, one of the incarnations of the quantum computers. We consider simpler models for which one can obtain exact mathemati- cal results. We prove existence of the dynamics in both Schroedinger and Heisenberg pictures, construct KMS states on appropriate C-algebras. We show (for high temperatures) that for each system where the lattice undergoes quantum evolution, there is a natural scaling leading to a quantum spin system on a fixed lattice Z, defined by a renormalized Hamiltonian

Probability Around the Quantum Gravity Part III.1: Planar Pure Gravity

Projet MEVALIn 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 $\alpha =-\frac{7}{2}$ exponent

Condensation in large closed Jackson networks

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Time Synchronization Model

There are two types i=1,2 of particles on the line R, with N_i particles of type i¸. Each particle of type i moves with constant velocity v_i. Moreover, any particle of type i=1,2 jumps to any particle of type j=1,2 with rates N_j^-1_ij. We find phase transitions in the clusterization (synchronization) behaviour of this system of particles on different time scales t=t(N) relative to N=N_1+N_2

Gibbs Measures on Attractors in Biological Neural Networks

Projet MEVALWe consider a class of nonreversible processes with a local interaction as a model of biological neural networks. We study the structure of equilibrium measures on attractors. There exist first and second order phase transitions with respect to the grey level parameter