Dimitri Petritis

Monte Carlo simulations of regularized bosonic strings

Abstract The world sheet of the bosonic string is regularized by triangulation. The corresponding statistical theory of triangulated random surfaces is simulated by Monte Carlo methods and the critical exponent γ for the susceptibility determined. A careful analysis of the systematic and statistical errors reveals that we have to wait for the next generation of computers before a high-precision measurement of γ is possible.

Extension of the Pirogov-Sinai theory to a class of quasiperiodic interactions

We extend the Pirogov-Sinai theory in such a manner that it applies to a large class of models with small quasiperiodic interactions as perturbations of periodic ones. We find general diophantine conditions on the frequency module of the quasiperiodic interactions and derivability conditions on the interaction potentials. These conditions allow to prove that the low temperature phase diagram is a homeomorphic deformation of the phase diagram at zero temperature.

Thermodynamic Formalism of Neural Computing

Neural networks are systems of interconnected processors mimicking some of the brain functions. After a rapid overview of neural computing, the thermodynamic formalism of the learning procedure is introduced. Besides its use in introducing efficient stochastic learning algorithms, it gives an insight in terms of information theory. Main emphasis is given in the information restitution process; stochastic evolution is used as the starting point for introducing statistical mechanics of associative memory. Instead of formulating problems in their most general setting, it is preferred stating precise results on specific models. In this report are mainly presented those features that are relevant when the neural net becomes very large. A survey of the most recent results is given and the main open problems are pointed out.

A Markov chain model of a polling system with parameter regeneration

We study a model of a polling system i.e.\ a collection of

Explosion, implosion, and moments of passage times for continuous-time Markov chains: a semimartingale approach

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Thermodynamics of a Brownian bridge polymer model in a random environment

queues with a single server that switches from queue to queue. The service time distribution and arrival rates change randomly every time a queue is emptied. This model is mapped to a mathematically equivalent model of a random walk with random choice of transition probabilities, a model which is of independent interest. All our results are obtained using methods from the constructive theory of Markov chains. We determine conditions for the existence of polynomial moments of hitting times for the random walk. An unusual phenomenon of thickness of the region of null recurrence for both the random walk and the queueing model is also proved.

Polling systems with parameter regeneration, the general case

We establish general theorems quantifying the notion of recurrence–through an estimation of the moments of passage times–for irreducible continuous-time Markov chains on countably infinite state spaces. Sharp conditions of occurrence of the phenomenon of explosion are also obtained. A new phenomenon of implosion is introduced and sharp conditions for its occurrence are proven. The general results are illustrated by treating models having a difficult behaviour even in discrete time.

Self-avoiding random walks on the hexagonal lattice

We consider a directed random walk making either 0 or +1 moves and a Brownian bridge, independent of the walk, conditioned to arrive at point b at time T. The Hamiltonian is defined as the sum of the square of the increments of the bridge between the moments of jump of the random walk and is interpreted as an energy function over the bridge configuration; the random walk acts as the random environment. This system provides a soluble model inspired by a closely related but distinct model of proteins. The thermodynamic limit of the quenched specific free energy is shown to exist and to be self-averaging, i.e. it is equal to a trivial - explicitly computed - random variable. An estimate of the asymptotic behaviour of the ground-state energy is also obtained.

Type transition of simple random walks on randomly directed regular lattices

We consider a polling model with multiple stations, each with Poisson arrivals and a queue of infinite capacity. The service regime is exhaustive and there is Jacksonian feedback of served customers. What is new here is that when the server comes to a station it chooses the service rate and the feedback parameters at random; these remain valid during the whole stay of the server at that station. We give criteria for recurrence, transience and existence of the

Semantic Distillation: A Method for Clustering Objects by their Contextual Specificity

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