16 research outputs found

    A Determinantal Formula for Catalan Tableaux and TASEP Probabilities

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    We present a determinantal formula for the steady state probability of each state of the TASEP (Totally Asymmetric Simple Exclusion Process) with open boundaries, a 1D particle model that has been studied extensively and displays rich combinatorial structure. These steady state probabilities are computed by the enumeration of Catalan tableaux, which are certain Young diagrams filled with α\alpha's and ÎČ\beta's that satisfy some conditions on the rows and columns. We construct a bijection from the Catalan tableaux to weighted lattice paths on a Young diagram, and from this we enumerate the paths with a determinantal formula, building upon a formula of Narayana that counts unweighted lattice paths on a Young diagram. Finally, we provide a formula for the enumeration of Catalan tableaux that satisfy a given condition on the rows, which corresponds to the steady state probability that in the TASEP on a lattice with nn sites, precisely kk of the sites are occupied by particles. This formula is an α / ÎČ\alpha\ /\ \beta generalization of the Narayana numbers.Comment: 19 pages, 12 figure

    Tableaux combinatorics for two-species PASEP probabilities

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    International audienceThe goal of this paper is to provide a combinatorial expression for the steady state probabilities of the twospecies PASEP. In this model, there are two species of particles, one “heavy” and one “light”, on a one-dimensional finite lattice with open boundaries. Both particles can hop into adjacent holes to the right and left at rates 1 and qq. Moreover, when the heavy and light particles are adjacent to each other, they can switch places as if the light particle were a hole. Additionally, the heavy particle can hop in and out at the boundary of the lattice. Our first result is a combinatorial interpretation for the stationary distribution at q=0q=0 in terms of certain multi-Catalan tableaux. We provide an explicit determinantal formula for the steady state probabilities, as well as some general enumerative results for this case. We also describe a Markov process on these tableaux that projects to the two-species PASEP, and hence directly explains the connection between the two. Finally, we extend our formula for the stationary distribution to the q=1q=1 case, using certain two-species alternative tableaux.Le but de ce document est de fournir une expression combinatoire dĂ©crivant les probabilitĂ©s de l’état d’équilibre de PASEP Ă  deux espĂšces. Dans ce modĂšle, il existe deux espĂšces de particules, une “lourde” et une “lĂ©gĂšre”, disposĂ©es sur un rĂ©seau fini unidimensionnel. Les deux particules peuvent sauter dans les trous adjacents Ă  droite et Ă  gauche, avec des probabilitĂ©s proportionnelles Ă  1 et qq. Par ailleurs, lorsque les particules lourdes et lĂ©gĂšres sont Ă  cĂŽtĂ© l’une de l’autre, elles peuvent changer de place, comme si la particule lĂ©gĂšre Ă©tait un trou. En outre, la particule lourde peut sauter dans et hors de la frontiĂšre du rĂ©seau. Notre premier rĂ©sultat est une interprĂ©tation combinatoire de la distribution stationnaire dans le cas q=0q=0, en termes de certains tableaux “multi-Catalan”. Nous proposons une formule explicite dĂ©terminantale pour les probabilitĂ©s stationnaires, ainsi que plusieurs rĂ©sultats Ă©numĂ©ratifs gĂ©nĂ©raux pour ce cas. Nous dĂ©crivons aussi un processus de Markov sur ces tableaux, qui se projette sur le PASEP Ă  deux espĂšces, et qui fournit donc directement une connexion entre les deux. Enfin, nous exprimons notre formule pour la distribution stationnaire dans le cas q=1q=1, en utilisant certains tableaux alternatifs de deux espĂšces

    Free fermionic probability theory and K-theoretic Schubert calculus

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    For each of the four particle processes given by Dieker and Warren [arXiv:0707.1843], we show the nn-step transition kernels are given by the (dual) (weak) refined symmetric Grothendieck functions up to a simple overall factor. We do so by encoding the particle dynamics as the basis of free fermions first introduced by the first author, which we translate into deformed Schur operators acting on partitions. We provide a direct combinatorial proof of this relationship in each case, where the defining tableaux naturally describe the particle motions.Comment: 52 pages, 5 figures, 2 table

    Combinatorial mappings of exclusion processes

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    We review various combinatorial interpretations and mappings of stationary-state probabilities of the totally asymmetric, partially asymmetric and symmetric simple exclusion processes (TASEP, PASEP, SSEP respectively). In these steady states, the statistical weight of a configuration is determined from a matrix product, which can be written explicitly in terms of generalised ladder operators. This lends a natural association to the enumeration of random walks with certain properties. Specifically, there is a one-to-many mapping of steady-state configurations to a larger state space of discrete paths, which themselves map to an even larger state space of number permutations. It is often the case that the configuration weights in the extended space are of a relatively simple form (e.g., a Boltzmann-like distribution). Meanwhile, various physical properties of the nonequilibrium steady state - such as the entropy - can be interpreted in terms of how this larger state space has been partitioned. These mappings sometimes allow physical results to be derived very simply, and conversely the physical approach allows some new combinatorial problems to be solved. This work brings together results and observations scattered in the combinatorics and statistical physics literature, and also presents new results. The review is pitched at statistical physicists who, though not professional combinatorialists, are competent and enthusiastic amateurs.Comment: 56 pages, 21 figure

    Nonequilibrium steady states from a random-walk perspective

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    It is well known that at thermal equilibrium (whereby a system has settled into a steady state with no energy or mass being exchanged with the environment), the microstates of a system are exponentially weighted by their energies, giving a Boltzmann distribution. All macroscopic quantities, such as the free energy and entropy, can be in principle computed given knowledge of the partition function. In a nonequilibrium steady state, on the other hand, the system has settled into a stationary state, but some currents of heat or mass persist. In the presence of these currents, there is no unified approach to solve for the microstate distribution. This motivates the central theme of this work, where I frame and solve problems in nonequilibrium statistical physics in terms of random walk and diffusion problems. The system that is the focus of Chapters 2, 3, and 4 is the (Totally) Asymmetric Simple Exclusion Process, or (T)ASEP. This is a system of hard-core particles making jumps through an open, one-dimensional lattice. This is a paradigmatic example of a nonequilibrium steady state that exhibits phase transitions. Furthermore, the probability of an arbitrary configuration of particles is exactly calculable, by a matrix product formalism that lends a natural association between the ASEP and a family of random walk problems. In Chapter 2 I present a unified description of the various combinatorial interpretations and mappings of steady-state configurations of the ASEP. As well as deriving new results, I bring together and unify results and observations that have otherwise been scattered in the combinatorics and physics literature. I show that particular particle configurations of the ASEP have a one-to-many mapping to a set of more abstract paths, which themselves have a one-to-many mapping to permutations of numbers. One observation from this wider literature has been that this mapped space can be interpreted as a larger set of configurations in some equilibrium system. This naturally gives an interpretation of ASEP configuration probabilities as summations of Boltzmann weights. The nonequilibrium partition function of the ASEP is then a summation over this equilibrium ensemble, however one encounters difficulties when calculating more detailed measures of this state space, such as the entropy. This motivates the work in Chapter 3. I calculate a quantity known as the RĂ©nyi entropy, which is a measure of the partitioning of the state space, and a deformation of the familiar Shannon entropy. The RĂ©nyi entropy is simple for an equilibrium system, but has yet to be explored in a classical nonequilibrium steady state. I use insights from Chapter 2 to frame one of these RĂ©nyi entropies | requiring the enumeration of the squares of configuration weights | in terms of a two-dimensional random walk with absorbing boundaries. I find the appropriate generating function across the full phase diagram of the TASEP by generalising a mathematical technique known as the obstinate kernel method. Importantly, this nonequilibrium RĂ©nyi entropy has a different structural form to any equilibrium system, highlighting a clear distinction between equilibrium and nonequilibrium distributions. In Chapter 4 I continue to examine the RĂ©nyi entropy of the TASEP, but now performing a time and space continuum limit of the random walk problem in Chapter 3. The resultant problem is a two-dimensional dffusion problem with absorbing boundary conditions, which once solved should recover TASEP dynamics about the point in the phase diagram where the three dynamical phases meet. I derive a generating function, sufficiently simple that its singularities can be analysed by hand. This calculation entails a novel generalisation of the obstinate kernel method of Chapter 3: I find a solution by exploiting a symmetry in the Laplace transform of the diffusion equation. I finish in Chapter 5 by introducing and solving another nonequilibrium system, termed the many-filament Brownian ratchet. This comprises an arbitrary number of filaments that stochastically grow and contract, with the net effect of moving a drift-diffusing membrane by purely from thermal fluctuations and steric interactions. These dynamics draw parallels with those of actin filament networks at the leading edge of eukaryotic cells, and this improves on previous 'pure ratchet' models by introducing interactions and heterogeneity in the filaments. I find an N-dimensional diffusion equation for the evolution of the N filament-membrane displacements. Several parameters can be varied in this system: the drift and diffusion rates of each of the filaments and membrane, the strength of a quadratic interaction between each filament with the membrane, and the strength of a surface tension across the filaments. For several interesting physical cases I find the steady-state distribution exactly, and calculate how the mean velocity of the membrane varies as a function of these parameters

    Mixing times for the TASEP in the maximal current phase

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    We study mixing times for the totally asymmetric simple exclusion process (TASEP) on a segment of size NN with open boundaries. We focus on the maximal current phase, and prove that the mixing time is of order N3/2N^{3/2}, up to logarithmic corrections. In the triple point, where the TASEP with open boundaries approaches the Uniform distribution on the state space, we show that the mixing time is precisely of order N3/2N^{3/2}. This is conjectured to be the correct order of the mixing time for a wide range of particle systems with maximal current. Our arguments rely on a connection to last-passage percolation, and recent results on moderate deviations of last-passage times.Comment: 42 pages, 10 figures, accepted versio
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