Luca Peliti

Multifractal wavefunction at the localisation threshold

The authors point out that the field theoretical description of the localisation transition exhibits anomalous scaling for the moments of the probability density at the localisation threshold. This property hints at a multifractal structure of the wavefunction. They attempt an analysis of the corresponding singularity distribution.

The Response of Glassy Systems to Random Perturbations: A Bridge Between Equilibrium and Off-Equilibrium

We discuss the response of aging systems with short-range interactions to a class of random perturbations. Although these systems are out of equilibrium, the limit value of the free energy at long times is equal to the equilibrium free energy. By exploiting this fact, we define a new order parameter function, and we relate it to the ratio between response and fluctuation, which is in principle measurable in an aging experiment. For a class of systems possessing stochastic stability, we show that this new order parameter function is intimately related to the static order parameter function, describing the distribution of overlaps between clustering states. The same method is applied to investigate the geometrical organization of pure states. We show that the ultrametric organization in the dynamics implies static ultrametricity, and we relate these properties to static separability, i.e., the property that the measure of the overlap between pure states is essentially unique. Our results, especially relevant for spin glasses, pave the way to an experimental determination of the order parameter function.

GEOGRAPHIC SPECIATION IN THE DERRIDA-HIGGS MODEL OF SPECIES FORMATION

We consider the Derrida-Higgs (DH) statistical model of species formation in the case where the population is geographically distributed in discrete locations, and mating only takes place within one location. Keeping the rate of migration between neighbouring locations at a fixed value, we change the mutation rate, changing therefore the average overlap between genotypes. When the overlap between individuals living in different locations falls below a fecundity threshold, speciation occurs. When more species coexist, the genetic structure of the population (as described by the overlap distribution P(q)) fluctuates. However, the average overlap, both within one location and among neighbouring locations, appears to vary according to the same laws as in the absence of speciation. The model provides a reasonable estimate of the parameter values necessary to observe geographic speciation, which is found to be much more likely than the sympatric speciation of the original DH model. Applications to the case of circular invasion, where the concept of biological species appears to run into difficulties, are sketched.

RANDOM-WALKS WITH MEMORY

Random walks with memory are a popular arena for methods aimed at describing irreversible aggregation phenomena. They produce topologically trivial aggregates whose embedding in ambient space can be described in terms of fractal concepts. The analogy with scaling properties of critical phenomena is well established and has led to a more complete understanding of their properties than for general aggregation processes.

Cross-over behavior of the nonlinear σ-model with quadratically broken symmetry☆

Abstract We study the nonlinear σ-model near two dimensions in the presence of a quadratic symmetry breaking, which gives a mass to M of the N fields. Using a renormalization scheme, proposed earlier, which includes the anisotropy mass explicitly, and making sure that the renormalized mass is a physical parameter, we calculate explicitly, to first order in d-2 : the flow patterns of the temperature (coupling constant); the crossover in temperature and external magnetic field of the order-parameter (the average classical field); and the cross-over in momentum of the two-point Green function.

Self-avoiding walks☆

Abstract The “true” self-avoiding walk is defined as the statistical problem of a traveller who steps randomly, but tries to avoid places he has already visited. The problem is contrasted to that of a chain with excluded volume, and is shown to be described by a class of Martin-Siggia-Rose field theories, a particular case of which represents some random walks in random environments. The results of renormalization-group calculations of the asymptotic behavior of these walks are reported.

On dangerous irrelevant operators

Abstract The effects of dangerous irrelevant operators on various types of critical behavior are described, as particular cases of a systematic field theoretic renormalization group treatment. Starting from a general formulation, such cases as the tricritical crossover above three dimensions, hyperscaling above four, and symmetry-breaking by irrelevant operators are considered. The irrelevance discussed is either oftthe “strong type”, identifiable by dimensional analysis, or of the “weak type”, produced by the renormalization group.

Heat Release by Controlled Continuous-Time Markov Jump Processes

We derive the equations governing the protocols minimizing the heat released by a continuous-time Markov jump process on a one-dimensional countable state space during a transition between assigned initial and final probability distributions in a finite time horizon. In particular, we identify the hypotheses on the transition rates under which the optimal control strategy and the probability distribution of the Markov jump problem obey a system of differential equations of Hamilton-Jacobi-Bellman-type. As the state-space mesh tends to zero, these equations converge to those satisfied by the diffusion process minimizing the heat released in the Langevin formulation of the same problem. We also show that in full analogy with the continuum case, heat minimization is equivalent to entropy production minimization. Thus, our results may be interpreted as a refined version of the second law of thermodynamics.

Population size effects in evolutionary dynamics on neutral networks and toy landscapes

We study the dynamics of a population subject to selective pressures, evolving either on RNA neutral networks or on toy fitness landscapes. We discuss the spread and the neutrality of the population in the steady state. Different limits arise depending on whether selection or random drift is dominant. In the presence of strong drift we show that the observables depend mainly on Mμ , M being the population size and μ the mutation rate, while corrections to this scaling go as 1/M : such corrections can be quite large in the presence of selection if there are barriers in the fitness landscape. Also we find that the convergence to the large- Mμ limit is linear in 1/M μ. Finally we introduce a protocol that minimizes drift; then observables scale like 1/M rather than 1/( Mμ ), allowing one to determine the large-M limit more quickly when μ is small; furthermore the genotypic diversity increases from O(ln M )t o O(M ).

Modeling microevolution in a changing environment: the evolving quasispecies and the diluted champion process

Several pathogens use evolvability as a survival strategy against acquired immunity of the host. Despite their high variability in time, some of them exhibit quite low variability within the population at any given time, a somewhat paradoxical behavior often called the evolving quasispecies. In this paper we introduce a simplified model of an evolving viral population in which the effects of the acquired immunity of the host are represented by the decrease of the fitness of the corresponding viral strains, depending on the frequency of the strain in the viral population. The model exhibits evolving quasispecies behavior in a certain range of its parameters, and suggests how punctuated evolution can be induced by a simple feedback mechanism.