We prove the invariance principle for a random Lorentz-gas particle in 3 dimensions under the Bol... more We prove the invariance principle for a random Lorentz-gas particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius r, placed according to a Poisson point process of density , in the limit → ∞, r → 0, r 2 → 1 up to time scales of order T = o(r −2 |log r| −2). To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (1969) [15, 16, 17], Spohn (1978) [27, 28] and Boldrighini-Bunimovich-Sinai (1983) [4]. The novelty is that the diffusive scaling of particle trajectory and the kinetic (Boltzmann-Grad) limit are taken simultaneously. The main ingredients are a coupling of the mechanical trajectory with the Markovian random flight process, and probabilistic and geometric controls on the efficiency of this coupling. Similar results have been earlier obtained for the weak coupling limit of classical and quantum random Lorentz gas, by Komorowski-Ryzhik (2006) [21], respectively, Erdős-Salmhofer-Yau (2007) [12, 13]. However, the following are substantial differences between our work and these ones: (1) The physical setting is different: low density rather than weak coupling. (2) The method of approach is different: probabilistic coupling rather than analytic/perturbative. (3) Due to (2), the time scale of validity of our diffusive approximation-expressed in terms of the kinetic time scale-is much longer and fully explicit.
We prove a central limit theorem under diffusive scaling for the displacement of a random walk on... more We prove a central limit theorem under diffusive scaling for the displacement of a random walk on Z d in stationary and ergodic doubly stochastic random environment, under the H −1-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result [10], where it is assumed that the stream tensor is in L max{2+δ,d} , with δ > 0. Our proof relies on an extension of the relaxed sector condition of [8], and is technically rather simpler than existing earlier proofs of similar results by Oelschläger [19] and Komorowski, Landim and Olla [10].
We survey recent results of normal and anomalous diffusion of two types of random motions with lo... more We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in R d or Z d. The first class consists of random walks on Z d in divergence-free random drift field, modelling the motion of a particle suspended in time-stationary incompressible turbulent flow. The second class consists of self-repelling random diffusions, where the diffusing particle is pushed by the negative gradient of its own occupation time measure towards regions less visited in the past. We establish normal diffusion (with square-rootof-time scaling and Gaussian limiting distribution) in three and more dimensions and typically anomalously fast diffusion in low dimensions (typically, one and two). Results are quoted from various papers published between 2012-2018, with some hints to the main ideas of the proofs. No technical details are presented here.
Consider a point particle moving through a Poisson distributed array of cubes all oriented along ... more Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes-the random wind-tree model introduced in Ehrenfest-Ehrenfest (1912) [6]. We show that, in the joint Boltzmann-Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to a Brownian motion in a particular scaling limit. In a previous paper (2019) [13] the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, the same strategy with some modification can be used to prove an invariance principle for the random wind-tree model.
Proceedings of the International Congress of Mathematicians (ICM 2018), 2019
We survey recent results of normal and anomalous diffusion of two types of random motions with lo... more We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in ${\Bbb R}^d$ or ${\Bbb Z}^d$. The first class consists of random walks on ${\Bbb Z}^d$ in divergence-free random drift field, modelling the motion of a particle suspended in time-stationary incompressible turbulent flow. The second class consists of self-repelling random diffusions, where the diffusing particle is pushed by the negative gradient of its own occupation time measure towards regions less visited in the past. We establish normal diffusion (with square-root-of-time scaling and Gaussian limiting distribution) in three and more dimensions and typically anomalously fast diffusion in low dimensions (typically, one and two). Results are quoted from various papers published between 2012-2018, with some hints to the main ideas of the proofs. No technical details are presented here.
Consider a point particle moving through a Poisson distributed array of cubes all oriented along ... more Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes-the random wind-tree model introduced in Ehrenfest-Ehrenfest (1912) [6]. We show that, in the joint Boltzmann-Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to a Brownian motion in a particular scaling limit. In a previous paper (2019) [13] the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, the same strategy with some modification can be used to prove an invariance principle for the random wind-tree model.
We prove the invariance principle for a random Lorentz-gas particle in 3 dimensions under the Bol... more We prove the invariance principle for a random Lorentz-gas particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius r, placed according to a Poisson point process of density $$\varrho $$ ϱ , in the limit $$\varrho \rightarrow \infty $$ ϱ → ∞ , $$r\rightarrow 0$$ r → 0 , $$\varrho r^{2}\rightarrow 1$$ ϱ r 2 → 1 up to time scales of order $$T=o(r^{-2}\left| {\log r}\right| ^{-2})$$ T = o ( r - 2 log r - 2 ) . To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (Phys Rev 185:308–322, 1969, Nota Interna Univ di Roma 358, 1970, Statistical mechanics. A short treatise. Theoretical and mathematical physics series, Springer, Berlin, 1999), Spohn (Commun Math Phys 60:277–290, 1978, Rev Mod Phys...
We prove the quenched version of the central limit theorem for the displacement of a random walk ... more We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the H −1-condition, with slightly stronger, L 2+ε (rather than L 2) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the nonreversible, divergence-free drift case, with unbounded (L 2+ε) stream tensor. This paper is a sequel of [Ann. Probab. 45 (2017) 4307-4347] and relies on technical results quoted from there.
We show that in any dimension $$d\ge 1$$ d ≥ 1 , the cycle-length process of stationary random st... more We show that in any dimension $$d\ge 1$$ d ≥ 1 , the cycle-length process of stationary random stirring (or, random interchange) on the lattice torus converges to the canonical Markovian split-and-merge process with the invariant (and reversible) measure given by the Poisson–Dirichlet law $$\mathsf {PD(1)}$$ PD ( 1 ) , as the size of the system grows to infinity. In the case of transient dimensions, $$d\ge 3$$ d ≥ 3 , the problem is motivated by attempts to understand the onset of long range order in quantum Heisenberg models via random loop representations of the latter.
We prove a central limit theorem under diffusive scaling for the displacement of a random walk on... more We prove a central limit theorem under diffusive scaling for the displacement of a random walk on Z d in stationary and ergodic doubly stochastic random environment, under the H −1-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result [Fluctuations in Markov Processes-Time Symmetry and Martingale Approximation (2012) Springer], where it is assumed that the stream tensor is in L max{2+δ,d} , with δ > 0. Our proof relies on an extension of the relaxed sector condition of [Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012) 463-476], and is technically rather simpler than existing earlier proofs of similar results by Oelschläger [Ann. Probab. 16 (1988) 1084-1126] and Komorowski, Landim and Olla [Fluctuations in Markov Processes-Time Symmetry and Martingale Approximation (2012) Springer].
In the randomly-oriented Manhattan lattice, every line in Z d is assigned a uniform random direct... more In the randomly-oriented Manhattan lattice, every line in Z d is assigned a uniform random direction. We consider the directed graph whose vertex set is Z d and whose edges connect nearest neighbours, but only in the direction fixed by the line orientations. Random walk on this directed graph chooses uniformly from the d legal neighbours at each step. We prove that this walk is superdiffusive in two and three dimensions. The model is diffusive in four and more dimensions.
This paper investigates the non-equilibrium hydrodynamic behavior of a simple totally asymmetric ... more This paper investigates the non-equilibrium hydrodynamic behavior of a simple totally asymmetric interacting particle system of particles, antiparticles and holes on Z. Rigorous hydrodynamic results apply to our model with a hydrodynamic flux that is exactly calculated and shown to change convexity in some region of the model parameters. We then characterize the entropy solutions of the hydrodynamic equation with step initial condition in this scenario which include various mixtures of rarefaction fans and shock waves. We highlight how the phase diagram of the model changes by varying the model parameters.
We consider a one-dimensional system consisting of a tagged particle of mass M surrounded by a ga... more We consider a one-dimensional system consisting of a tagged particle of mass M surrounded by a gas of unit-mass hard-point particles in thermal equilibrium. Denoting by Q t the displacement of the tagged particle, we give Q 2 lower and upper bounds -independent of M -for lim£-. It results from the proof that the correct nontrivial norming of Q t -if any -is j/ί.
1Mathematical Institute of the Hungarian Academy of Sciences, POB 127, H-1364 Budapest, Hungary (... more 1Mathematical Institute of the Hungarian Academy of Sciences, POB 127, H-1364 Budapest, Hungary (e-mail: [email protected]) 2CNRS, Laboratoire de Mathématiques, ENS, 45, rue d'Ulm, F-75230 Paris cedex 05, France (e-mail: [email protected])
We investigate the asymptotic properties of a random tree growth model which generalizes the basi... more We investigate the asymptotic properties of a random tree growth model which generalizes the basic concept of preferential attachment. The Barabási-Albert random graph model is based on the idea that the popularity of a vertex in the graph (the probability that a new ...
We survey the current status of the list of questions related to the favourite (or: most visited)... more We survey the current status of the list of questions related to the favourite (or: most visited) sites of simple random walk on Z, raised by Pál Erd˝os and Pál Révész in the early eighties. ... Let (S(n), n ∈ Z+) be a simple symmetric random walk on Z with S(0) = 0. Define
In the present paper we continue the investigation of the so-called coalescing ideal gas in one d... more In the present paper we continue the investigation of the so-called coalescing ideal gas in one dimension, initiated by Ermakov. The model consists of point-like particles moving with velocities±1 which coalesce and choose a fresh velocity with the same distribution when ...
The study of 1d Brownian trajectories pushed up or down by Skorohod-reflection on some other Borw... more The study of 1d Brownian trajectories pushed up or down by Skorohod-reflection on some other Borwnian trajectories (running backwards in time) was initiated in [5] and motivated in [7] by the construction of the object what is today called the Brownian Web, see [3]. It turns out that ...
1.1. Historical background. Let X(t), t ∈ Z+ := {0, 1, 2,...} be a nearest neigh-bour walk on the... more 1.1. Historical background. Let X(t), t ∈ Z+ := {0, 1, 2,...} be a nearest neigh-bour walk on the integer lattice Z starting from X(0) = 0 and denote by ℓ(t, x), (t, x) ∈ Z+ × Z, its local time (that is: its occupation time measure) on sites: ℓ(t, x) := #{0 ≤ s ≤ t : X(s) = x} where #{...} ...
We prove the invariance principle for a random Lorentz-gas particle in 3 dimensions under the Bol... more We prove the invariance principle for a random Lorentz-gas particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius r, placed according to a Poisson point process of density , in the limit → ∞, r → 0, r 2 → 1 up to time scales of order T = o(r −2 |log r| −2). To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (1969) [15, 16, 17], Spohn (1978) [27, 28] and Boldrighini-Bunimovich-Sinai (1983) [4]. The novelty is that the diffusive scaling of particle trajectory and the kinetic (Boltzmann-Grad) limit are taken simultaneously. The main ingredients are a coupling of the mechanical trajectory with the Markovian random flight process, and probabilistic and geometric controls on the efficiency of this coupling. Similar results have been earlier obtained for the weak coupling limit of classical and quantum random Lorentz gas, by Komorowski-Ryzhik (2006) [21], respectively, Erdős-Salmhofer-Yau (2007) [12, 13]. However, the following are substantial differences between our work and these ones: (1) The physical setting is different: low density rather than weak coupling. (2) The method of approach is different: probabilistic coupling rather than analytic/perturbative. (3) Due to (2), the time scale of validity of our diffusive approximation-expressed in terms of the kinetic time scale-is much longer and fully explicit.
We prove a central limit theorem under diffusive scaling for the displacement of a random walk on... more We prove a central limit theorem under diffusive scaling for the displacement of a random walk on Z d in stationary and ergodic doubly stochastic random environment, under the H −1-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result [10], where it is assumed that the stream tensor is in L max{2+δ,d} , with δ > 0. Our proof relies on an extension of the relaxed sector condition of [8], and is technically rather simpler than existing earlier proofs of similar results by Oelschläger [19] and Komorowski, Landim and Olla [10].
We survey recent results of normal and anomalous diffusion of two types of random motions with lo... more We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in R d or Z d. The first class consists of random walks on Z d in divergence-free random drift field, modelling the motion of a particle suspended in time-stationary incompressible turbulent flow. The second class consists of self-repelling random diffusions, where the diffusing particle is pushed by the negative gradient of its own occupation time measure towards regions less visited in the past. We establish normal diffusion (with square-rootof-time scaling and Gaussian limiting distribution) in three and more dimensions and typically anomalously fast diffusion in low dimensions (typically, one and two). Results are quoted from various papers published between 2012-2018, with some hints to the main ideas of the proofs. No technical details are presented here.
Consider a point particle moving through a Poisson distributed array of cubes all oriented along ... more Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes-the random wind-tree model introduced in Ehrenfest-Ehrenfest (1912) [6]. We show that, in the joint Boltzmann-Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to a Brownian motion in a particular scaling limit. In a previous paper (2019) [13] the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, the same strategy with some modification can be used to prove an invariance principle for the random wind-tree model.
Proceedings of the International Congress of Mathematicians (ICM 2018), 2019
We survey recent results of normal and anomalous diffusion of two types of random motions with lo... more We survey recent results of normal and anomalous diffusion of two types of random motions with long memory in ${\Bbb R}^d$ or ${\Bbb Z}^d$. The first class consists of random walks on ${\Bbb Z}^d$ in divergence-free random drift field, modelling the motion of a particle suspended in time-stationary incompressible turbulent flow. The second class consists of self-repelling random diffusions, where the diffusing particle is pushed by the negative gradient of its own occupation time measure towards regions less visited in the past. We establish normal diffusion (with square-root-of-time scaling and Gaussian limiting distribution) in three and more dimensions and typically anomalously fast diffusion in low dimensions (typically, one and two). Results are quoted from various papers published between 2012-2018, with some hints to the main ideas of the proofs. No technical details are presented here.
Consider a point particle moving through a Poisson distributed array of cubes all oriented along ... more Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes-the random wind-tree model introduced in Ehrenfest-Ehrenfest (1912) [6]. We show that, in the joint Boltzmann-Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to a Brownian motion in a particular scaling limit. In a previous paper (2019) [13] the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, the same strategy with some modification can be used to prove an invariance principle for the random wind-tree model.
We prove the invariance principle for a random Lorentz-gas particle in 3 dimensions under the Bol... more We prove the invariance principle for a random Lorentz-gas particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius r, placed according to a Poisson point process of density $$\varrho $$ ϱ , in the limit $$\varrho \rightarrow \infty $$ ϱ → ∞ , $$r\rightarrow 0$$ r → 0 , $$\varrho r^{2}\rightarrow 1$$ ϱ r 2 → 1 up to time scales of order $$T=o(r^{-2}\left| {\log r}\right| ^{-2})$$ T = o ( r - 2 log r - 2 ) . To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (Phys Rev 185:308–322, 1969, Nota Interna Univ di Roma 358, 1970, Statistical mechanics. A short treatise. Theoretical and mathematical physics series, Springer, Berlin, 1999), Spohn (Commun Math Phys 60:277–290, 1978, Rev Mod Phys...
We prove the quenched version of the central limit theorem for the displacement of a random walk ... more We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the H −1-condition, with slightly stronger, L 2+ε (rather than L 2) integrability condition on the stream tensor. On the way we extend Nash's moment bound to the nonreversible, divergence-free drift case, with unbounded (L 2+ε) stream tensor. This paper is a sequel of [Ann. Probab. 45 (2017) 4307-4347] and relies on technical results quoted from there.
We show that in any dimension $$d\ge 1$$ d ≥ 1 , the cycle-length process of stationary random st... more We show that in any dimension $$d\ge 1$$ d ≥ 1 , the cycle-length process of stationary random stirring (or, random interchange) on the lattice torus converges to the canonical Markovian split-and-merge process with the invariant (and reversible) measure given by the Poisson–Dirichlet law $$\mathsf {PD(1)}$$ PD ( 1 ) , as the size of the system grows to infinity. In the case of transient dimensions, $$d\ge 3$$ d ≥ 3 , the problem is motivated by attempts to understand the onset of long range order in quantum Heisenberg models via random loop representations of the latter.
We prove a central limit theorem under diffusive scaling for the displacement of a random walk on... more We prove a central limit theorem under diffusive scaling for the displacement of a random walk on Z d in stationary and ergodic doubly stochastic random environment, under the H −1-condition imposed on the drift field. The condition is equivalent to assuming that the stream tensor of the drift field be stationary and square integrable. This improves the best existing result [Fluctuations in Markov Processes-Time Symmetry and Martingale Approximation (2012) Springer], where it is assumed that the stream tensor is in L max{2+δ,d} , with δ > 0. Our proof relies on an extension of the relaxed sector condition of [Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012) 463-476], and is technically rather simpler than existing earlier proofs of similar results by Oelschläger [Ann. Probab. 16 (1988) 1084-1126] and Komorowski, Landim and Olla [Fluctuations in Markov Processes-Time Symmetry and Martingale Approximation (2012) Springer].
In the randomly-oriented Manhattan lattice, every line in Z d is assigned a uniform random direct... more In the randomly-oriented Manhattan lattice, every line in Z d is assigned a uniform random direction. We consider the directed graph whose vertex set is Z d and whose edges connect nearest neighbours, but only in the direction fixed by the line orientations. Random walk on this directed graph chooses uniformly from the d legal neighbours at each step. We prove that this walk is superdiffusive in two and three dimensions. The model is diffusive in four and more dimensions.
This paper investigates the non-equilibrium hydrodynamic behavior of a simple totally asymmetric ... more This paper investigates the non-equilibrium hydrodynamic behavior of a simple totally asymmetric interacting particle system of particles, antiparticles and holes on Z. Rigorous hydrodynamic results apply to our model with a hydrodynamic flux that is exactly calculated and shown to change convexity in some region of the model parameters. We then characterize the entropy solutions of the hydrodynamic equation with step initial condition in this scenario which include various mixtures of rarefaction fans and shock waves. We highlight how the phase diagram of the model changes by varying the model parameters.
We consider a one-dimensional system consisting of a tagged particle of mass M surrounded by a ga... more We consider a one-dimensional system consisting of a tagged particle of mass M surrounded by a gas of unit-mass hard-point particles in thermal equilibrium. Denoting by Q t the displacement of the tagged particle, we give Q 2 lower and upper bounds -independent of M -for lim£-. It results from the proof that the correct nontrivial norming of Q t -if any -is j/ί.
1Mathematical Institute of the Hungarian Academy of Sciences, POB 127, H-1364 Budapest, Hungary (... more 1Mathematical Institute of the Hungarian Academy of Sciences, POB 127, H-1364 Budapest, Hungary (e-mail: [email protected]) 2CNRS, Laboratoire de Mathématiques, ENS, 45, rue d'Ulm, F-75230 Paris cedex 05, France (e-mail: [email protected])
We investigate the asymptotic properties of a random tree growth model which generalizes the basi... more We investigate the asymptotic properties of a random tree growth model which generalizes the basic concept of preferential attachment. The Barabási-Albert random graph model is based on the idea that the popularity of a vertex in the graph (the probability that a new ...
We survey the current status of the list of questions related to the favourite (or: most visited)... more We survey the current status of the list of questions related to the favourite (or: most visited) sites of simple random walk on Z, raised by Pál Erd˝os and Pál Révész in the early eighties. ... Let (S(n), n ∈ Z+) be a simple symmetric random walk on Z with S(0) = 0. Define
In the present paper we continue the investigation of the so-called coalescing ideal gas in one d... more In the present paper we continue the investigation of the so-called coalescing ideal gas in one dimension, initiated by Ermakov. The model consists of point-like particles moving with velocities±1 which coalesce and choose a fresh velocity with the same distribution when ...
The study of 1d Brownian trajectories pushed up or down by Skorohod-reflection on some other Borw... more The study of 1d Brownian trajectories pushed up or down by Skorohod-reflection on some other Borwnian trajectories (running backwards in time) was initiated in [5] and motivated in [7] by the construction of the object what is today called the Brownian Web, see [3]. It turns out that ...
1.1. Historical background. Let X(t), t ∈ Z+ := {0, 1, 2,...} be a nearest neigh-bour walk on the... more 1.1. Historical background. Let X(t), t ∈ Z+ := {0, 1, 2,...} be a nearest neigh-bour walk on the integer lattice Z starting from X(0) = 0 and denote by ℓ(t, x), (t, x) ∈ Z+ × Z, its local time (that is: its occupation time measure) on sites: ℓ(t, x) := #{0 ≤ s ≤ t : X(s) = x} where #{...} ...
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