Sequential and asynchronous processes driven by stochastic or quantum grammars and their application to genomics: a survey

We present the formalism of sequential and asynchronous processes defined in terms of random or quantum grammars and argue that these processes have relevance in genomics. To make the article accessible to the non-mathematicians, we keep the mathematical exposition as elementary as possible, focusing on some general ideas behind the formalism and stating the implications of the known mathematical results. We close with a set of open challenging problems.Comment: Presented at the European Congress on Mathematical and Theoretical Biology, Dresden 18--22 July 200

On the physical relevance of random walks: an example of random walks on a randomly oriented lattice

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 general undirected graphs are associated with (not necessarily Abelian) groups while directed graphs are associated with (not necessarily Abelian) $C^*$-algebras. Since quantum mechanics is naturally formulated in terms of $C^*$-algebras, the study of random walks on directed lattices has been motivated lately by the development of the new field of quantum information and communication

Equilibrium statistical mechanics of frustrated spin glasses; a survey of mathematical results

After a rapid introduction to the physical motivations and a succinct presentation of heuristic results, this survey summarises the main mathematical results known on the Edwards-Anderson and the Sherrington-Kirkpatrick models of spin glasses. Although not complete proofs but rather sketches of the relevant steps and important ideas are given, only results for which complete proofs are known --- and for which the author has been able to reproduce all the intermediate logical steps --- are presented in the sections entitled `mathematical results'. This paper is intended to both physicists, interested to know which articles among the multitude of papers published on the subject go beyond the heuristic arguments to obtain rigorous irrefutable results, but also to the mathematicians, interested in finding out how rich is the physical intuitive way of thinking and in being inspired by the heuristic results in view of a mathematical rigorisation. An extended, but not exhaustive, bibliography is included.Comment: 29 pages, postscrip

Random environment on coloured trees

In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this model and also show how our model generalizes many other probabilistic models, including random walk in random environment on trees, recursive distributional equations and multi-type branching random walk on $\mathbb{R}$.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ101 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Bindweeds or random walks in random environments on multiplexed trees and their asympotics

We report on the asymptotic behaviour of a new model of random walk, we term the bindweed model, evolving in a random environment on an infinite multiplexed tree. The term \textit{multiplexed} means that the model can be viewed as a nearest neighbours random walk on a tree whose vertices carry an internal degree of freedom from the finite set $\{1,...,d\}$, for some integer $d$. The consequence of the internal degree of freedom is an enhancement of the tree graph structure induced by the replacement of ordinary edges by multi-edges, indexed by the set $\{1,...,d\}\times\{1,...,d\}$. This indexing conveys the information on the internal degree of freedom of the vertices contiguous to each edge. The term \textit{random environment} means that the jumping rates for the random walk are a family of edge-indexed random variables, independent of the natural filtration generated by the random variables entering in the definition of the random walk; their joint distribution depends on the index of each component of the multi-edges. We study the large time asymptotic behaviour of this random walk and classify it with respect to positive recurrence or transience in terms of a specific parameter of the probability distribution of the jump rates. This classifying parameter is shown to coincide with the critical value of a matrix-valued multiplicative cascade on the ordinary tree (\textit{i.e.} the one without internal degrees of freedom attached to the vertices) having the same vertex set as the state space of the random walk. Only results are presented here since the detailed proofs will appear elsewhere

Dynamical systems with heavy-tailed random parameters

Motivated by the study of the time evolution of random dynamical systems arising in a vast variety of domains --- ranging from physics to ecology ---, we establish conditions for the occurrence of a non-trivial asymptotic behaviour for these systems in the absence of an ellipticity condition. More precisely, we classify these systems according to their type and --- in the recurrent case --- provide with sharp conditions quantifying the nature of recurrence by establishing which moments of passage times exist and which do not exist. The problem is tackled by mapping the random dynamical systems into Markov chains on $\mathbb{R}$ with heavy-tailed innovation and then using powerful methods stemming from Lyapunov functions to map the resulting Markov chains into positive semi-martingales.Comment: 24 page

THERMODYNAMICS OF A BROWNIAN BRIDGE POLYMER MODEL IN A RANDOM ENVIRONMENT

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 $b$ on time $T$. The Hamiltonian is defined as the sum of the square of increments of the bridge between the moments of jump of the random walk and interpreted as an energy function over the bridge connfiguration; the random walk acts as the random environment. This model provides a continuum version of a model with some relevance to protein conformation. The thermodynamic limit of the 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.Comment: 20 pages, uuencoded postscrip

On the pertinence to Physics of random walks induced by random dynamical systems: a survey

International audienceLet $\mathbb{X}$ be an abstract space and $\mathbb{A}$ a denumerable (finite or infinite) alphabet. Suppose that $(p_a)_{a\in\mathbb{A}}$ is a family of functions $p_a:\mathbb{X}\to\mathbb{R}_+$ such that for all $x\in\mathbb{X}$ we have $\sum_{a\in\mathbb{A}} p_a(x)=1$ and $(S_a)_{a\in\mathbb{A}}$ a family of transformations $S_a:\mathbb{X}\to\mathbb{X}$. The pair $((S_a)_a, (p_a)_a)$ is termed an \textit{iterated function system with place dependent probabilities}. Such systems can be thought as generalisations of random dynamical systems. As a matter of fact, suppose we start from a given $x\in\mathbb{X}$; we pick then randomly, with probability $p_a(x)$, \text{the} transformation $S_a$ and evolve to $S_a(x)$. We are interested in the behaviour of the system when the iteration continues indefinitely.Random walks of the above type are omnipresent in both classical and quantum Physics. To give a small sample of occurrences we mention: random walks on the affine group, random walks on Penrose lattices, random walks on partially directed lattices, evolution of density matrices induced by repeated quantum measurements, quantum channels, quantum random walks, etc.In this article, we review some basic properties of such systems and provide with a pathfinder in the extensive bibliography (both on mathematical and physical sides) where the main results have been originally published