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13
QuasiStationary Regime of a Branching Random Walk in Presence of an Absorbing Wall
 J STAT PHYS
, 2008
"... A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as the velocity v of the wall varies. Below the critical velocity vc, the population has a nonzero survival probability and when the population survives its size grows exponential ..."
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A branching random walk in presence of an absorbing wall moving at a constant velocity v undergoes a phase transition as the velocity v of the wall varies. Below the critical velocity vc, the population has a nonzero survival probability and when the population survives its size grows exponentially. We investigate the histories of the population conditioned on having a single survivor at some final time T. We study the quasistationary regime for v < vc when T is large. To do so, one can construct a modified stochastic process which is equivalent to the original process conditioned on having a single survivor at final time T. We then use this construction to show that the properties of the quasistationary regime are universal when v → vc. We also solve exactly a simple version of the problem, the exponential model, for which the study of the quasistationary regime can be reduced to the analysis of a single onedimensional map.
Effects of nonlocal feedback on traveling fronts in neural fields subject to transmission delay
, 2004
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A stochastic model for wound healing
, 2008
"... We present a discrete stochastic model which represents many of the salient features of the biological process of wound healing. The model describes fronts of cells invading a wound. We have numerical results in one and two dimensions. In one dimension we can give analytic results for the front spee ..."
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We present a discrete stochastic model which represents many of the salient features of the biological process of wound healing. The model describes fronts of cells invading a wound. We have numerical results in one and two dimensions. In one dimension we can give analytic results for the front speed as a power series expansion in a parameter, p, that gives the relative size of proliferation and diffusion processes for the invading cells. In two dimensions the model becomes the Eden model for p≈1. In both one and two dimensions for small p, front propagation for this model should approach that of the FisherKolmogorov equation. However, as in other cases, this discrete model approaches FisherKolmogorov behavior slowly. 1
Quantum chromodynamics at high energy and statistical physics
, 901
"... When hadrons are scattered at high energies, strong color fields, whose dynamics is described by quantum chromodynamics (QCD), are generated at the interaction point. If one represents these fields in terms of partons (quarks and gluons), the average number densities of the latter saturate at ultrah ..."
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When hadrons are scattered at high energies, strong color fields, whose dynamics is described by quantum chromodynamics (QCD), are generated at the interaction point. If one represents these fields in terms of partons (quarks and gluons), the average number densities of the latter saturate at ultrahigh energies. At that point, nonlinear effects become predominant in the dynamical equations. The hadronic states that one gets in this regime of QCD are generically called “color glass condensates”. Our understanding of scattering in QCD has benefited from recent progress in statistical and mathematical physics. The evolution of hadronic scattering amplitudes at fixed impact parameter in the regime where nonlinear parton saturation effects become sizable was shown to be similar to the time evolution of a system of classical particles undergoing reactiondiffusion processes. The dynamics of such a system is essentially governed by equations in the universality class of the stochastic FisherKolmogorovPetrovskyPiscounov equation, which is a stochastic nonlinear partial differential equation. Realizations of that kind of equations (that is, “events” in a particle physics language) have the form of noisy traveling waves. Universal properties of the latter can be taken over to scattering amplitudes in QCD. This review provides an introduction to the basic methods of statistical physics useful in QCD, and summarizes the correspondence between these two fields and its theoretical and phenomenological implications. Key words: Quantum chromodynamics, color dipole model, color glass condensate, stochastic fronts, traveling waves, reactiondiffusion
LARGE DEVIATIONS OF THE FRONT IN A ONE DIMENSIONAL MODEL OF X + Y → 2X
"... We investigate the probabilities of large deviations for the position of the front in a stochastic model of the reaction X + Y → 2X on the integer lattice in which Y particles do not move while X particles move as independent simple continuous time random walks of total jump rate 2. For a wide class ..."
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We investigate the probabilities of large deviations for the position of the front in a stochastic model of the reaction X + Y → 2X on the integer lattice in which Y particles do not move while X particles move as independent simple continuous time random walks of total jump rate 2. For a wide class of initial conditions, we prove that a large deviations principle holds and we show that the zero set of the rate function is the interval [0, v], where v is the velocity of the front given by the law of large numbers. We also give more precise estimates for the rate of decay of the slowdown probabilities. Our results indicate a gapless property of the generator of the process as seen from the front, as it happens in the context of nonlinear diffusion equations describing the propagation of a pulled front into an unstable state.
FLUCTUATIONS OF THE FRONT IN A ONE DIMENSIONAL MODEL FOR THE SPREAD OF AN INFECTION
"... We study the following microscopic model of infection or epidemic reaction: red and blue particles perform independent continuoustime nearestneighbor symmetric random walks on the integer lattice Z with jump rates DR for red particles and DB for blue particles, the interaction rule being that blue ..."
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We study the following microscopic model of infection or epidemic reaction: red and blue particles perform independent continuoustime nearestneighbor symmetric random walks on the integer lattice Z with jump rates DR for red particles and DB for blue particles, the interaction rule being that blue particles turn red upon contact with a red particle. The initial condition consists of i.i.d. Poisson particle numbers at each site, with particles at the left of the origin being red, while particles at the right of the origin are blue. We are interested in the dynamics of the front, defined as the rightmost position of a red particle. For the case DR = DB (in fact, for a general d−dimensional version of it), Kesten and Sidoravicius established that the front moves ballistically, and more precisely that it satisfies a law of large numbers. In this paper, we prove that a central limit theorem for the front holds when DR = DB. Moreover, this result can be extended to the case where DR> DB, up to modifying the dynamics so that blue particles turn red upon contact with a site that has previously been occupied by a red particle. Our approach is based on the definition of a renewal structure, extending ideas developed by Comets, Quastel and Ramírez for the socalled frog model, where DB = 0.
Front velocity and directed polymers in random medium
 Stochastic Processes and their Applications
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LARGE DEVIATIONS OF THE FRONT IN A ONE DIMENSIONAL MODEL OF X + Y → 2X
, 2008
"... Abstract. We investigate the probabilities of large deviations for the position of the front in a stochastic model of the reaction X + Y → 2X on the integer lattice in which Y particles do not move while X particles move as independent simple continuous time random walks of total jump rate 2. For a ..."
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Abstract. We investigate the probabilities of large deviations for the position of the front in a stochastic model of the reaction X + Y → 2X on the integer lattice in which Y particles do not move while X particles move as independent simple continuous time random walks of total jump rate 2. For a wide class of initial conditions, we prove that a large deviations principle holds and we show that the zero set of the rate function is the interval [0, v], where v is the velocity of the front given by the law of large numbers. We also give more precise estimates for the rate of decay of the slowdown probabilities. Our results indicate a gapless property of the generator of the process as seen from the front, as it happens in the context of nonlinear diffusion equations describing the propagation of a pulled front into an unstable state. 1.
Asymptotic Shape and Propagation of Fronts for Growth Models in Dynamic Random Environment
"... We survey recent rigorous results and open problems related to models of Interacting Particle Systems which describe the autocatalytic type reaction A + B → 2B, with diffusion constants of particles being respectively DA ≥ 0 and DB ≥ 0. Depending on the choice of the values of DA and DB, we cover ..."
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We survey recent rigorous results and open problems related to models of Interacting Particle Systems which describe the autocatalytic type reaction A + B → 2B, with diffusion constants of particles being respectively DA ≥ 0 and DB ≥ 0. Depending on the choice of the values of DA and DB, we cover three distinct cases: so called ”rumor or infection spread ” model (DA> 0, DB> 0); the Stochastic Combustion process (DA = 0 and DB> 0); and finally the ”modified ” Diffusion Limited Aggregation, which corresponds to the case DA> 0, DB = 0 with modified transition rule: A + B → 2B occurs when A and Bparticles become nearest neighbors and Aparticle attempts to jump on a vertex where Bparticle is located. Then such jump is suppressed, and Aparticle becomes Bparticle.