arXiv subjects
Publications and source records attributed to Raphaël Cerf.
We study Eigen's quasispecies model in the asymptotic regime where the length of the genotypes goes to infinity and the mutation probability goes to 0. We give several explicit formulas for the stationary solutions of the limiting system of differential equations.
We try to design a simple model exhibiting self-organized criticality, which is amenable to a rigorous mathematical analysis. To this end, we modify the generalized Ising Curie-Weiss model by implementing an automatic control of the inverse temperature. For a class of symmetric distributions whose density satisfies some integrability conditions, we prove that the sum $S_n$ of the random variables behaves as in the typical critical generalized Ising Curie-Weiss model. The fluctuations are of order $n^{3/4}$, and the limiting law is $C\exp(-λx^4)\,dx$ where $C$ and $λ$ are suitable positive constants.
We introduce a new parameter to discuss the behavior of a genetic algorithm. This parameter is the mean number of exact copies of the best fit chromosomes from one generation to the next. We argue that the genetic algorithm should operate efficiently when this parameter is slightly larger than $1$. We consider the case of the simple genetic algorithm with the roulette--wheel selection mechanism. We denote by $\ell$ the length of the chromosomes, by $m$ the population size, by $p_C$ the crossover probability and by $p_M$ the mutation probability. We start the genetic algorithm with an initial population whose maximal fitness is equal to $f_0^*$ and whose mean fitness is equal to ${\overline{f_0}}$. We show that, in the limit of large populations, the dynamics of the genetic algorithm depends in a critical way on the parameter $π\,=\,\big({f_0^*}/{\overline{f_0}}\big) (1-p_C)(1-p_M)^\ell\,.$ Our results suggest that the mutation and crossover probabilities should be tuned so that, at each generation, $\text{maximal fitness} \times (1-p_C) (1-p_M)^\ell > \text{mean fitness}$.
We consider the standard site percolation model on the $d$-dimensional lattice. A direct consequence of the proof of the uniqueness of the infinite cluster of Aizenman, Kesten and Newman [Comm. Math. Phys. 111 (1987) 505-531] is that the two-arms exponent is larger than or equal to $1/2$. We improve slightly this lower bound in any dimension $d\geq2$. Next, starting only with the hypothesis that $θ(p)>0$, without using the slab technology, we derive a quantitative estimate establishing long-range order in a finite box.
We pursue the task of developing a finite population counterpart to Eigen's model. We consider the classical Wright-Fisher model describing the evolution of a population of size $m$ of chromosomes of length $\ell$ over an alphabet of cardinality $κ$. The mutation probability per locus is $q$. The replication rate is $σ>1$ for the master sequence and $1$ for the other sequences. We study the equilibrium distribution of the process in the regime where $\ell\to+\infty$, $m\to+\infty$, $q\to0$, $\ell q\to a\in\,]0,+\infty[$, $\frac{m}{\ell}\toα\in [0,+\infty]$. We obtain an equation $αψ(a)=\lnκ$ in the parameter space $(a,α)$ separating the regime where the equilibrium population is totally random from the regime where a quasispecies is formed. We observe the existence of a critical population size necessary for a quasispecies to emerge, and we recover the finite population counterpart of the error threshold. The result is the twin brother of the corresponding result for the Moran model. The proof is more complex, and it relies on the Freidlin-Wentzell theory of random perturbations of dynamical systems.
We consider the model of i.i.d. first passage percolation on $\mathbb{Z}^d$ : we associate with each edge $e$ of the graph a passage time $t(e)$ taking values in $[0,+\infty]$, such that $\mathbb{P}[t(e)<+\infty] >p_c(d)$. Equivalently, we consider a standard (finite) i.i.d. first passage percolation model on a super-critical Bernoulli percolation performed independently. We prove a weak shape theorem without any moment assumption. We also prove that the corresponding time constant is positive if and only if $\mathbb{P}[t(e)=0]<p_c(d)$.
Let $ρ$ and $μ$ be two probability measures on $\mathbb{R}$ which are not the Dirac mass at $0$. We denote by $H(μ|ρ)$ the relative entropy of $μ$ with respect to $ρ$. We prove that, if $ρ$ is symmetric and $μ$ has a finite first moment, then \[ H(μ|ρ)\geq \frac{\displaystyle{(\int_{\mathbb{R}}z\,dμ(z))^2}}{\displaystyle{2\int_{\mathbb{R}}z^2\,dμ(z)}}\,,\] with equality if and only if $μ=ρ$.
We prove the following exponential inequality: Let $n\geq 1$ and let $X_1,...,X_n$ be $n$ independent identically distributed symmetric real-valued random variables. For any $x,y>0$, we have \[\mathbb{P}\big({X_1+...+X_n}\geq x,\, {X_1^2+...+X_n^2}\leq y\big)< \exp(-\frac{x^2}{2y})\,.\]
We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq2$ and a domain $Ω$ of boundary $Γ$ in $\mathbb{R}^d$. Let $Γ^1$ and $Γ^2$ be two disjoint open subsets of $Γ$, representing the parts of $Γ$ through which some water can enter and escape from $Ω$. A law of large numbers for the maximal flow from $Γ^1$ to $Γ^2$ in $Ω$ is already known. In this paper we investigate the asymptotic behavior of a maximal stream and a minimal cutset. A maximal stream is a vector measure $\vecμ_n^{\max}$ that describes how the maximal amount of fluid can cross $Ω$. Under conditions on the regularity of the domain and on the law of the capacities of the edges, we prove that the sequence $(\vecμ_n^{\max})_{n\geq1}$ converges a.s. to the set of the solutions of a continuous deterministic problem of maximal stream in an anisotropic network. A minimal cutset can been seen as the boundary of a set $E_n^{\min}$ that separates $Γ^1$ from $Γ^2$ in $Ω$ and whose random capacity is minimal. Under the same conditions, we prove that the sequence $(E_n^{\min})_{n\geq1}$ converges toward the set of the solutions of a continuous deterministic problem of minimal cutset. We deduce from this a continuous deterministic max-flow min-cut theorem and a new proof of the law of large numbers for the maximal flow. This proof is more natural than the existing one, since it relies on the study of maximal streams and minimal cutsets, which are the pertinent objects to look at.
We study the simple genetic algorithm with a ranking selection mechanism (linear ranking or tournament). We denote by $\ell$ the length of the chromosomes, by $m$ the population size, by $p_C$ the crossover probability and by $p_M$ the mutation probability. We introduce a parameter $σ$, called the selection drift, which measures the selection intensity of the fittest chromosome. We show that the dynamics of the genetic algorithm depend in a critical way on the parameter $$π\,=\,σ(1-p_C)(1-p_M)^\ell\,.$$ If $π<1$, then the genetic algorithm operates in a disordered regime: an advantageous mutant disappears with probability larger than $1-1/m^β$, where $β$ is a positive exponent. If $π>1$, then the genetic algorithm operates in a quasispecies regime: an advantageous mutant invades a positive fraction of the population with probability larger than a constant $p^*$ (which does not depend on $m$). We estimate next the probability of the occurrence of a catastrophe (the whole population falls below a fitness level which was previously reached by a positive fraction of the population). The asymptotic results suggest the following rules: $π=σ(1-p_C)(1-p_M)^\ell$ should be slightly larger than $1$; $p_M$ should be of order $1/\ell$; $m$ should be larger than $\ell\ln\ell$; the running time should be of exponential order in $m$. The first condition requires that $ \ell p_M +p_C< \lnσ$. These conclusions must be taken with great care: they come from an asymptotic regime, and it is a formidable task to understand the relevance of this regime for a real-world problem. At least, we hope that these conclusions provide interesting guidelines for the practical implementation of the simple genetic algorithm.
This work extends to dimension $d\geq3$ the main result of Dehghanpour and Schonmann. We consider the stochastic Ising model on ${\mathbb{Z}}^d$ evolving with the Metropolis dynamics under a fixed small positive magnetic field $h$ starting from the minus phase. When the inverse temperature $β$ goes to $\infty$, the relaxation time of the system, defined as the time when the plus phase has invaded the origin, behaves like $\exp({βκ_d})$. The value $κ_d$ is equal to \[κ_d=\frac{1}{d+1}(Γ_1+\cdots+Γ_d),\] where $Γ_i$ is the energy of the $i$-dimensional critical droplet of the Ising model at zero temperature and magnetic field $h$.
We consider the Moran model on the sharp peak landscape, in the asymptotic regime studied in [3], where a quasispecies is formed. We find explicitly the distribution of this quasispecies.
We consider the standard site percolation model on the three dimensional cubic lattice. Starting solely with the hypothesis that $θ(p)>0$, we prove that, for any $α>0$, there exists $κ>0$ such that, with probability larger than $1-1/n^α$, every pair of vertices inside the box $Λ(n)$ are joined by a path having at most $κ(\ln n)^2$ closed sites.
The goal of this work is to propose a finite population counterpart to Eigen's model, which incorporates stochastic effects. We consider a Moran model describing the evolution of a population of size $m$ of chromosomes of length $\ell$ over an alphabet of cardinality $κ$. The mutation probability per locus is $q$. We deal only with the sharp peak landscape: the replication rate is $σ>1$ for the master sequence and 1 for the other sequences. We study the equilibrium distribution of the process in the regime where $\ell, m\to +\infty$, $q\to 0$, $\ell q \to a$, $m/\ell\toα$. We obtain an equation $αϕ(a)=\lnκ$ in the parameter space $(a,α)$ separating the regime where the equilibrium population is totally random from the regime where a quasispecies is formed. We observe the existence of a critical population size necessary for a quasispecies to emerge and we recover the finite population counterpart of the error threshold. These results are supported by computer simulations.
We consider the standard first passage percolation model in the rescaled graph $\mathbb {Z}^d/n$ for $d\geq2$ and a domain $Ω$ of boundary $Γ$ in $\mathbb {R}^d$. Let $Γ^1$ and $Γ^2$ be two disjoint open subsets of $Γ$ representing the parts of $Γ$ through which some water can enter and escape from $Ω$. We investigate the asymptotic behavior of the flow $ϕ_n$ through a discrete version $Ω_n$ of $Ω$ between the corresponding discrete sets $Γ^1_n$ and $Γ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the upper large deviations of $ϕ_n/n^{d-1}$ above a certain constant are of volume order, that is, decays exponentially fast with $n^d$. This article is part of a larger project in which the authors prove that this constant is the a.s. limit of $ϕ_n/n^{d-1}$.
We give a general setting for Cramér's large deviations theorem for the empirical means of a field of random vectors, which contains Cramér's theorem for i.i.d. random vectors and Sanov's theorem for asymptotically decoupled measures. ----- Nous établissons un cadre général pour le théorème de Cramér sur les grandes déviations des moyennes empiriques d'un champ de vecteurs aléatoires, cadre qui contient le théorème de Cramér pour des vecteurs aléatoires i.i.d. et le théorème de Sanov pour les mesures asymptotiquement découplées.
Based on speculations coming from statistical mechanics and the conjectured existence of critical states, I propose a simple heuristic in order to control the mutation probability and the population size of a genetic algorithm.
We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $Ω$ of boundary $Γ$ in $\mathbb{R}^d$. Let $Γ^1$ and $Γ^2$ be two disjoint open subsets of $Γ$, representing the parts of $Γ$ through which some water can enter and escape from $Ω$. We investigate the asymptotic behaviour of the flow $ϕ_n$ through a discrete version $Ω_n$ of $Ω$ between the corresponding discrete sets $Γ^1_n$ and $Γ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviations of $ϕ_n/ n^{d-1}$ below a certain constant are of surface order.