DOI

1

Finite size corrections in the random energy model and the replica approach - Derrida, Bernard and Mottishaw, Peter

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT , (2015)

Abstract : We present a systematic and exact way of computing finite size corrections for the random energy model, in its low temperature phase. We obtain explicit (though complicated) expressions for the finite size corrections of the overlap functions. In its low temperature phase, the random energy model is known to exhibit Parisi's broken symmetry of replicas. The finite size corrections given by our exact calculation can be reproduced using replicas if we make specific assumptions about the fluctuations (with negative variances!) of the number and sizes of the blocks when replica symmetry is broken. As an alternative we show that the exact expression for the non-integer moments of the partition function can be written in terms of coupled contour integrals over what can be thought of as `complex replica numbers'. Parisi's one step replica symmetry breaking arises naturally from the saddle point of these integrals without making any ansatz or using the replica method. The fluctuations of the `complex replica numbers' near the saddle point in the imaginary direction correspond to the negative variances we observed in the replica calculation. Finally our approach allows one to see why some apparently diverging series or integrals are harmless.

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT , (2015)

LPS

Abstract : We present a systematic and exact way of computing finite size corrections for the random energy model, in its low temperature phase. We obtain explicit (though complicated) expressions for the finite size corrections of the overlap functions. In its low temperature phase, the random energy model is known to exhibit Parisi's broken symmetry of replicas. The finite size corrections given by our exact calculation can be reproduced using replicas if we make specific assumptions about the fluctuations (with negative variances!) of the number and sizes of the blocks when replica symmetry is broken. As an alternative we show that the exact expression for the non-integer moments of the partition function can be written in terms of coupled contour integrals over what can be thought of as `complex replica numbers'. Parisi's one step replica symmetry breaking arises naturally from the saddle point of these integrals without making any ansatz or using the replica method. The fluctuations of the `complex replica numbers' near the saddle point in the imaginary direction correspond to the negative variances we observed in the replica calculation. Finally our approach allows one to see why some apparently diverging series or integrals are harmless.

DOI

2

Large deviation function of a tracer position in single file diffusion - Sadhu, Tridib and Derrida, Bernard

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT , (2015)

Abstract : Diffusion of impenetrable particles in a crowded one-dimensional channel is referred as the single file diffusion. The particles do not pass each other and the displacement of each individual particle is sub-diffusive. We analyse a simple realization of this single file diffusion problem where one dimensional Brownian point particles interact only by hard-core repulsion. We show that the large deviation function which characterizes the displacement of a tracer at large time can be computed via a mapping to a problem of non-interacting Brownian particles. We confirm recently obtained results of the one time distribution of the displacement and show how to extend them to the multi-time correlations. The probability distribution of the tracer position depends on whether we take annealed or quenched averages. In the quenched case we notice an exact relation between the distribution of the tracer and the distribution of the current. This relation is in fact much more general and would be valid for arbitrary single file diffusion. It allows in particular to get the full statistics of the tracer position for the symmetric simple exclusion process (SSEP) at density 1/2 in the quenched case.

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT , (2015)

LPS

Abstract : Diffusion of impenetrable particles in a crowded one-dimensional channel is referred as the single file diffusion. The particles do not pass each other and the displacement of each individual particle is sub-diffusive. We analyse a simple realization of this single file diffusion problem where one dimensional Brownian point particles interact only by hard-core repulsion. We show that the large deviation function which characterizes the displacement of a tracer at large time can be computed via a mapping to a problem of non-interacting Brownian particles. We confirm recently obtained results of the one time distribution of the displacement and show how to extend them to the multi-time correlations. The probability distribution of the tracer position depends on whether we take annealed or quenched averages. In the quenched case we notice an exact relation between the distribution of the tracer and the distribution of the current. This relation is in fact much more general and would be valid for arbitrary single file diffusion. It allows in particular to get the full statistics of the tracer position for the symmetric simple exclusion process (SSEP) at density 1/2 in the quenched case.

DOI

3

Log-periodic Critical Amplitudes: A Perturbative Approach - Derrida, Bernard and Giacomin, Giambattista

JOURNAL OF STATISTICAL PHYSICS 154, 286-304 (2014)

Abstract : Log-periodic amplitudes appear in the critical behavior of a large class of systems, in particular when a discrete scale invariance is present. Here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial. In this case, the log-periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated.

JOURNAL OF STATISTICAL PHYSICS 154, 286-304 (2014)

LPS

Abstract : Log-periodic amplitudes appear in the critical behavior of a large class of systems, in particular when a discrete scale invariance is present. Here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial. In this case, the log-periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated.

DOI

4

The Depinning Transition in Presence of Disorder: A Toy Model - Derrida, Bernard and Retaux, Martin

JOURNAL OF STATISTICAL PHYSICS 156, 268-290 (2014)

Abstract : We introduce a toy model, which represents a simplified version of the problem of the depinning transition in the limit of strong disorder. This toy model can be formulated as a simple renormalization transformation for the probability distribution of a single real variable. For this toy model, the critical line is known exactly in one particular case and it can be calculated perturbatively in the general case. One can also show that, at the transition, there is no fixed distribution accessible by renormalization which corresponds to a disordered fixed point. Instead, both our numerical and analytic approaches indicate a transition of infinite order (of the Berezinskii-Kosterlitz-Thouless type). We give numerical evidence that this infinite order transition persists for the problem of the depinning transition with disorder on the hierarchical lattice.

JOURNAL OF STATISTICAL PHYSICS 156, 268-290 (2014)

LPS

Abstract : We introduce a toy model, which represents a simplified version of the problem of the depinning transition in the limit of strong disorder. This toy model can be formulated as a simple renormalization transformation for the probability distribution of a single real variable. For this toy model, the critical line is known exactly in one particular case and it can be calculated perturbatively in the general case. One can also show that, at the transition, there is no fixed distribution accessible by renormalization which corresponds to a disordered fixed point. Instead, both our numerical and analytic approaches indicate a transition of infinite order (of the Berezinskii-Kosterlitz-Thouless type). We give numerical evidence that this infinite order transition persists for the problem of the depinning transition with disorder on the hierarchical lattice.

DOI

5

Genealogies in simple models of evolution - Brunet, Eric and Derrida, Bernard

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT , (2013)

Abstract : We review the statistical properties of the genealogies of a few models of evolution. In the asexual case, selection leads to coalescence times which grow logarithmically with the size of the population, in contrast with the linear growth of the neutral case. Moreover for a whole class of models, the statistics of the genealogies are those of the Bolthausen-Sznitman coalescent rather than the Kingman coalescent in the neutral case. For sexual reproduction in the neutral case, the time to reach the first common ancestors for the whole population and the time for all individuals to have all their ancestors in common are also logarithmic in the population size, as predicted by Chang in 1999. We discuss how these times are modified by introducing selection in a simple way.

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT , (2013)

LPS

Abstract : We review the statistical properties of the genealogies of a few models of evolution. In the asexual case, selection leads to coalescence times which grow logarithmically with the size of the population, in contrast with the linear growth of the neutral case. Moreover for a whole class of models, the statistics of the genealogies are those of the Bolthausen-Sznitman coalescent rather than the Kingman coalescent in the neutral case. For sexual reproduction in the neutral case, the time to reach the first common ancestors for the whole population and the time for all individuals to have all their ancestors in common are also logarithmic in the population size, as predicted by Chang in 1999. We discuss how these times are modified by introducing selection in a simple way.

DOI

6

Exact solution of a Levy walk model for anomalous heat transport - Dhar, Abhishek and Saito, Keiji and Derrida, Bernard

PHYSICAL REVIEW E 87, (2013)

Abstract : The Levy walk model is studied in the context of the anomalous heat conduction of one-dimensional systems. In this model, the heat carriers execute Levy walks instead of normal diffusion as expected in systems where Fourier's law holds. Here we calculate exactly the average heat current, the large deviation function of its fluctuations, and the temperature profile of the Levy walk model maintained in a steady state by contact with two heat baths (the open geometry). We find that the current is nonlocally connected to the temperature gradient. As observed in recent simulations of mechanical models, all the cumulants of the current fluctuations have the same system-size dependence in the open geometry. For the ring geometry, we argue that a size-dependent cutoff time is necessary for the Levy walk model to behave like mechanical models. This modification does not affect the results on transport in the open geometry for large enough system sizes. DOI: 10.1103/PhysRevE.87.010103

PHYSICAL REVIEW E 87, (2013)

LPS

Abstract : The Levy walk model is studied in the context of the anomalous heat conduction of one-dimensional systems. In this model, the heat carriers execute Levy walks instead of normal diffusion as expected in systems where Fourier's law holds. Here we calculate exactly the average heat current, the large deviation function of its fluctuations, and the temperature profile of the Levy walk model maintained in a steady state by contact with two heat baths (the open geometry). We find that the current is nonlocally connected to the temperature gradient. As observed in recent simulations of mechanical models, all the cumulants of the current fluctuations have the same system-size dependence in the open geometry. For the ring geometry, we argue that a size-dependent cutoff time is necessary for the Levy walk model to behave like mechanical models. This modification does not affect the results on transport in the open geometry for large enough system sizes. DOI: 10.1103/PhysRevE.87.010103

DOI

7

Universal current fluctuations in the symmetric exclusion process and other diffusive systems - Akkermans, Eric and Bodineau, Thierry and Derrida, Bernard and Shpielberg, Ohad

EPL 103, (2013)

Abstract : Using the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim, one can show that the statistics of the current of the symmetric simple exclusion process (SSEP) connected to two reservoirs on an arbitrary large finite domain in dimension d are the same as in the one-dimensional case. Numerical results on squares support this claim while results on cubes exhibit some discrepancy. We argue that the results of the macroscopic fluctuation theory should be recovered by increasing the size of the contacts. The generalization to other diffusive systems is straightforward. Copyright (C) EPLA, 2013

EPL 103, (2013)

LPS

Abstract : Using the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim, one can show that the statistics of the current of the symmetric simple exclusion process (SSEP) connected to two reservoirs on an arbitrary large finite domain in dimension d are the same as in the one-dimensional case. Numerical results on squares support this claim while results on cubes exhibit some discrepancy. We argue that the results of the macroscopic fluctuation theory should be recovered by increasing the size of the contacts. The generalization to other diffusive systems is straightforward. Copyright (C) EPLA, 2013

DOI

8

Finite Size Corrections to the Large Deviation Function of the Density in the One Dimensional Symmetric Simple Exclusion Process - Derrida, Bernard and Retaux, Martin

JOURNAL OF STATISTICAL PHYSICS 152, 824-845 (2013)

Abstract : The symmetric simple exclusion process is one of the simplest out-of-equilibrium systems for which the steady state is known. Its large deviation functional of the density has been computed in the past both by microscopic and macroscopic approaches. Here we obtain the leading finite size correction to this large deviation functional. The result is compared to the similar corrections for equilibrium systems.

JOURNAL OF STATISTICAL PHYSICS 152, 824-845 (2013)

LPS

Abstract : The symmetric simple exclusion process is one of the simplest out-of-equilibrium systems for which the steady state is known. Its large deviation functional of the density has been computed in the past both by microscopic and macroscopic approaches. Here we obtain the leading finite size correction to this large deviation functional. The result is compared to the similar corrections for equilibrium systems.

DOI

9

How genealogies are affected by the speed of evolution - Brunet, Eric and Derrida, Bernard

PHILOSOPHICAL MAGAZINE 92, 255-271 (2012)

Abstract : In a series of recent works it has been shown that a class of simple models of evolving populations under selection leads to genealogical trees whose statistics are given by the Bolthausen-Sznitman coalescent rather than by the well-known Kingman coalescent in the case of neutral evolution. Here we show that when conditioning the genealogies on the speed of evolution, one finds a one-parameter family of tree statistics which interpolates between the Bolthausen-Sznitman and Kingman coalescents. This interpolation can be calculated explicitly for one specific version of the model, the exponential model. Numerical simulations of another version of the model and a phenomenological theory indicate that this one-parameter family of tree statistics could be universal. We compare this tree structure with those appearing in other contexts, in particular in the mean field theory of spin glasses.

PHILOSOPHICAL MAGAZINE 92, 255-271 (2012)

LPS

Abstract : In a series of recent works it has been shown that a class of simple models of evolving populations under selection leads to genealogical trees whose statistics are given by the Bolthausen-Sznitman coalescent rather than by the well-known Kingman coalescent in the case of neutral evolution. Here we show that when conditioning the genealogies on the speed of evolution, one finds a one-parameter family of tree statistics which interpolates between the Bolthausen-Sznitman and Kingman coalescents. This interpolation can be calculated explicitly for one specific version of the model, the exponential model. Numerical simulations of another version of the model and a phenomenological theory indicate that this one-parameter family of tree statistics could be universal. We compare this tree structure with those appearing in other contexts, in particular in the mean field theory of spin glasses.

DOI

10

Anomalous long-range correlations at a non-equilibrium phase transition - Gerschenfeld, A. and Derrida, B.

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 45, (2012)

Abstract : Non-equilibrium diffusive systems are known to exhibit long-range correlations, which decay like the inverse 1/L of the system size L in one dimension. Here, taking the example of the ABC model, we show that this size dependence becomes anomalous (the decay becomes a non-integer power of L) when the diffusive system approaches a second-order phase transition. This power-law decay as well as the L-dependence of the time-time correlations can be understood in terms of the dynamics of the amplitude of the first Fourier mode of the particle densities. This amplitude evolves according to a Langevin equation in a quartic potential, which was introduced in a previous work to explain the anomalous behavior of the cumulants of the current near this second-order phase transition. Here we also compute some of these cumulants away from the transition and show that they become singular as the transition is approached, matching with what we already knew in the critical regime.

JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 45, (2012)

LPS

Abstract : Non-equilibrium diffusive systems are known to exhibit long-range correlations, which decay like the inverse 1/L of the system size L in one dimension. Here, taking the example of the ABC model, we show that this size dependence becomes anomalous (the decay becomes a non-integer power of L) when the diffusive system approaches a second-order phase transition. This power-law decay as well as the L-dependence of the time-time correlations can be understood in terms of the dynamics of the amplitude of the first Fourier mode of the particle densities. This amplitude evolves according to a Langevin equation in a quartic potential, which was introduced in a previous work to explain the anomalous behavior of the cumulants of the current near this second-order phase transition. Here we also compute some of these cumulants away from the transition and show that they become singular as the transition is approached, matching with what we already knew in the critical regime.

DOI

11

Microscopic versus macroscopic approaches to non-equilibrium systems - Derrida, Bernard

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT , (2011)

Abstract : The one-dimensional symmetric simple exclusion process (SSEP) is one of the very few exactly soluble models of non-equilibrium statistical physics. It describes a system of particles which diffuse with hard core repulsion on a one-dimensional lattice in contact with two reservoirs of particles at unequal densities. The goal of this paper is to review the two main approaches which lead to the exact expression of the large deviation functional of the density of the SSEP in its steady state: a microscopic approach (based on the matrix product ansatz and an additivity property) and a macroscopic approach (based on the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim).

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT , (2011)

LPS

Abstract : The one-dimensional symmetric simple exclusion process (SSEP) is one of the very few exactly soluble models of non-equilibrium statistical physics. It describes a system of particles which diffuse with hard core repulsion on a one-dimensional lattice in contact with two reservoirs of particles at unequal densities. The goal of this paper is to review the two main approaches which lead to the exact expression of the large deviation functional of the density of the SSEP in its steady state: a microscopic approach (based on the matrix product ansatz and an additivity property) and a macroscopic approach (based on the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim).

DOI

12

A Branching Random Walk Seen from the Tip - Brunet, Eric and Derrida, Bernard

JOURNAL OF STATISTICAL PHYSICS 143, 420-446 (2011)

Abstract : We show that all the time-dependent statistical properties of the rightmost points of a branching Brownian motion can be extracted from the traveling wave solutions of the Fisher-KPP equation. The distribution of all the distances between the rightmost points has a long time limit which can be understood as the delay of the Fisher-KPP traveling waves when the initial condition is modified. The limiting measure exhibits the surprising property of superposability: the statistical properties of the distances between the rightmost points of the union of two realizations of the branching Brownian motion shifted by arbitrary amounts are the same as those of a single realization. We discuss the extension of our results to more general branching random walks.

JOURNAL OF STATISTICAL PHYSICS 143, 420-446 (2011)

LPS

Abstract : We show that all the time-dependent statistical properties of the rightmost points of a branching Brownian motion can be extracted from the traveling wave solutions of the Fisher-KPP equation. The distribution of all the distances between the rightmost points has a long time limit which can be understood as the delay of the Fisher-KPP traveling waves when the initial condition is modified. The limiting measure exhibits the surprising property of superposability: the statistical properties of the distances between the rightmost points of the union of two realizations of the branching Brownian motion shifted by arbitrary amounts are the same as those of a single realization. We discuss the extension of our results to more general branching random walks.

DOI

13

Current fluctuations at a phase transition - Gerschenfeld, A. and Derrida, B.

EPL 96, (2011)

Abstract : The ABC model is a simple diffusive one-dimensional non-equilibrium system which exhibits a phase transition. Here we show that the cumulants of the currents of particles through the system become singular near the phase transition. At the transition, they exhibit an anomalous dependence on the system size (an anomalous Fourier's law). An effective theory for the dynamics of the single mode which becomes unstable at the transition allows one to predict this anomalous scaling. Copyright (C) EPLA, 2011

EPL 96, (2011)

LPS

Abstract : The ABC model is a simple diffusive one-dimensional non-equilibrium system which exhibits a phase transition. Here we show that the cumulants of the currents of particles through the system become singular near the phase transition. At the transition, they exhibit an anomalous dependence on the system size (an anomalous Fourier's law). An effective theory for the dynamics of the single mode which becomes unstable at the transition allows one to predict this anomalous scaling. Copyright (C) EPLA, 2011

DOI

14

Phase Fluctuations in the ABC Model - Bodineau, T. and Derrida, B.

JOURNAL OF STATISTICAL PHYSICS 145, 745-762 (2011)

Abstract : We analyze the fluctuations of the steady state profiles in the modulated phase of the ABC model. For a system of L sites, the steady state profiles move on a microscopic time scale of order L (3). The variance of their displacement is computed in terms of the macroscopic steady state profiles by using fluctuating hydrodynamics and large deviations. Our analytical prediction for this variance is confirmed by the results of numerical simulations.

JOURNAL OF STATISTICAL PHYSICS 145, 745-762 (2011)

LPS

Abstract : We analyze the fluctuations of the steady state profiles in the modulated phase of the ABC model. For a system of L sites, the steady state profiles move on a microscopic time scale of order L (3). The variance of their displacement is computed in terms of the macroscopic steady state profiles by using fluctuating hydrodynamics and large deviations. Our analytical prediction for this variance is confirmed by the results of numerical simulations.

DOI

15

Fluctuations of the heat flux of a one-dimensional hard particle gas - Brunet, E. and Derrida, B. and Gerschenfeld, A.

EPL 90, (2010)

Abstract : Momentum-conserving one-dimensional models are known to exhibit anomalous Fourier's law, with a thermal conductivity varying as a power law of the system size. Here we measure, by numerical simulations, several cumulants of the heat flux of a one-dimensional hard particle gas. We find that the cumulants, like the conductivity, vary as power laws of the system size. Our results also indicate that cumulants higher than the second follow different power laws when one compares the ring geometry at equilibrium and the linear case in contact with two heat baths (at equal or unequal temperatures). Copyright (C) EPLA, 2010

EPL 90, (2010)

LPS

Abstract : Momentum-conserving one-dimensional models are known to exhibit anomalous Fourier's law, with a thermal conductivity varying as a power law of the system size. Here we measure, by numerical simulations, several cumulants of the heat flux of a one-dimensional hard particle gas. We find that the cumulants, like the conductivity, vary as power laws of the system size. Our results also indicate that cumulants higher than the second follow different power laws when one compares the ring geometry at equilibrium and the linear case in contact with two heat baths (at equal or unequal temperatures). Copyright (C) EPLA, 2010

DOI

16

A Diffusive System Driven by a Battery or by a Smoothly Varying Field - Bodineau, T. and Derrida, B. and Lebowitz, J. L.

JOURNAL OF STATISTICAL PHYSICS 140, 648-675 (2010)

Abstract : We consider the steady state of a one dimensional diffusive system, such as the symmetric simple exclusion process (SSEP) on a ring, driven by a battery at the origin or by a smoothly varying field along the ring. The battery appears as the limiting case of a smoothly varying field, when the field becomes a delta function at the origin. We find that in the scaling limit the long range pair correlation functions of the system driven by a battery are very different from the ones known in the steady state of the SSEP maintained out of equilibrium by contact with two reservoirs, even when the steady state density profiles are identical in both models.

JOURNAL OF STATISTICAL PHYSICS 140, 648-675 (2010)

LPS

Abstract : We consider the steady state of a one dimensional diffusive system, such as the symmetric simple exclusion process (SSEP) on a ring, driven by a battery at the origin or by a smoothly varying field along the ring. The battery appears as the limiting case of a smoothly varying field, when the field becomes a delta function at the origin. We find that in the scaling limit the long range pair correlation functions of the system driven by a battery are very different from the ones known in the steady state of the SSEP maintained out of equilibrium by contact with two reservoirs, even when the steady state density profiles are identical in both models.

DOI

17

Anomalous Fourier's Law and Long Range Correlations in a 1D Non-momentum Conserving Mechanical Model - Gerschenfeld, A. and Derrida, B. and Lebowitz, J. L.

JOURNAL OF STATISTICAL PHYSICS 141, 757-766 (2010)

Abstract : We study by means of numerical simulations the velocity reversal model, a one-dimensional mechanical model of heat transport introduced in 1985 by Ianiro and Lebowitz. Our numerical results indicate that this model, which does not conserve momentum, exhibits nevertheless an anomalous Fourier's law similar to the ones previously observed in momentum-conserving models. This disagrees with what can be expected by solving the Boltzmann equation (BE) for this system. The pair correlation velocity field also looks very different from the correlations usually seen in diffusive systems, and shares some similarity with those of momentum-conserving heat transport models.

JOURNAL OF STATISTICAL PHYSICS 141, 757-766 (2010)

LPS

Abstract : We study by means of numerical simulations the velocity reversal model, a one-dimensional mechanical model of heat transport introduced in 1985 by Ianiro and Lebowitz. Our numerical results indicate that this model, which does not conserve momentum, exhibits nevertheless an anomalous Fourier's law similar to the ones previously observed in momentum-conserving models. This disagrees with what can be expected by solving the Boltzmann equation (BE) for this system. The pair correlation velocity field also looks very different from the correlations usually seen in diffusive systems, and shares some similarity with those of momentum-conserving heat transport models.

DOI

18

Fractional Moment Bounds and Disorder Relevance for Pinning Models - Derrida, Bernard and Giacomin, Giambattista and Lacoin, Hubert and Toninelli, Fabio Lucio

COMMUNICATIONS IN MATHEMATICAL PHYSICS 287, 867-887 (2009)

Abstract : We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(center dot) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n) = n(-alpha-1) L(n), with alpha >= 0 and L(center dot) slowly varying. The model undergoes a (de)-localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For alpha < 1/2 disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents [3,28]. The same has been proven also for alpha = 1/2, but under the assumption that L(center dot) diverges sufficiently fast at infinity, a hypothesis that is not satisfied in the (1 + 1)-dimensional wetting model considered in [12,17], where L(center dot) is asymptotically constant. Here we prove that, if 1/2 < alpha < 1 or alpha > 1, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the so-called Harris criterion, disorder is therefore relevant in this case. In the marginal case alpha = 1/2, under the assumption that L(center dot) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed critical points, which is smaller than any power of the disorder strength, is positive: disorder is marginally relevant. Again, the case considered in [12,17] is out of our analysis and remains open. The results are achieved by setting the parameters of the model so that the annealed system is localized, but close to criticality, and by first considering a quenched system of size that does not exceed the correlation length of the annealed model. In such a regime we can show that the expectation of the partition function raised to a suitably chosen power gamma is an element of (0, 1) is small. We then exploit such an information to prove that the expectation of the same fractional power of the partition function goes to zero with the size of the system, a fact that immediately entails that the quenched system is delocalized.

COMMUNICATIONS IN MATHEMATICAL PHYSICS 287, 867-887 (2009)

LPS

Abstract : We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(center dot) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n) = n(-alpha-1) L(n), with alpha >= 0 and L(center dot) slowly varying. The model undergoes a (de)-localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For alpha < 1/2 disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents [3,28]. The same has been proven also for alpha = 1/2, but under the assumption that L(center dot) diverges sufficiently fast at infinity, a hypothesis that is not satisfied in the (1 + 1)-dimensional wetting model considered in [12,17], where L(center dot) is asymptotically constant. Here we prove that, if 1/2 < alpha < 1 or alpha > 1, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the so-called Harris criterion, disorder is therefore relevant in this case. In the marginal case alpha = 1/2, under the assumption that L(center dot) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed critical points, which is smaller than any power of the disorder strength, is positive: disorder is marginally relevant. Again, the case considered in [12,17] is out of our analysis and remains open. The results are achieved by setting the parameters of the model so that the annealed system is localized, but close to criticality, and by first considering a quenched system of size that does not exceed the correlation length of the annealed model. In such a regime we can show that the expectation of the partition function raised to a suitably chosen power gamma is an element of (0, 1) is small. We then exploit such an information to prove that the expectation of the same fractional power of the partition function goes to zero with the size of the system, a fact that immediately entails that the quenched system is delocalized.

DOI

19

Current Fluctuations of the One Dimensional Symmetric Simple Exclusion Process with Step Initial Condition - Derrida, Bernard and Gerschenfeld, Antoine

JOURNAL OF STATISTICAL PHYSICS 136, 1-15 (2009)

Abstract : For the symmetric simple exclusion process on an infinite line, we calculate exactly the fluctuations of the integrated current Q(t) during time t through the origin when, in the initial condition, the sites are occupied with density rho(a) on the negative axis and with density rho(b) on the positive axis. All the cumulants of Q(t) grow like root t. In the range where Q(t) similar to root t, the decay exp[-Q(t)(3)/t] of the distribution of Q(t) is non-Gaussian. Our results are obtained using the Bethe ansatz and several identities derived recently by Tracy and Widom for exclusion processes on the infinite line.

JOURNAL OF STATISTICAL PHYSICS 136, 1-15 (2009)

LPS

Abstract : For the symmetric simple exclusion process on an infinite line, we calculate exactly the fluctuations of the integrated current Q(t) during time t through the origin when, in the initial condition, the sites are occupied with density rho(a) on the negative axis and with density rho(b) on the positive axis. All the cumulants of Q(t) grow like root t. In the range where Q(t) similar to root t, the decay exp[-Q(t)(3)/t] of the distribution of Q(t) is non-Gaussian. Our results are obtained using the Bethe ansatz and several identities derived recently by Tracy and Widom for exclusion processes on the infinite line.

DOI

20

Statistics at the tip of a branching random walk and the delay of traveling waves - Brunet, E. and Derrida, B.

EPL 87, (2009)

Abstract : We study the limiting distribution of particles at the frontier of a branching random walk. The positions of these particles can be viewed as the lowest energies of a directed polymer in a random medium in the mean-field case. We show that the average distances between these leading particles can be computed as the delay of a traveling wave evolving according to the Fisher-KPP front equation. These average distances exhibit universal behaviors, different from those of the probability cascades studied recently in the context of mean-field spin-glasses. Copyright (C) EPLA, 2009

EPL 87, (2009)

LPS

Abstract : We study the limiting distribution of particles at the frontier of a branching random walk. The positions of these particles can be viewed as the lowest energies of a directed polymer in a random medium in the mean-field case. We show that the average distances between these leading particles can be computed as the delay of a traveling wave evolving according to the Fisher-KPP front equation. These average distances exhibit universal behaviors, different from those of the probability cascades studied recently in the context of mean-field spin-glasses. Copyright (C) EPLA, 2009