16 research outputs found
A Determinantal Formula for Catalan Tableaux and TASEP Probabilities
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 's and '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 sites, precisely of the sites are occupied by particles.
This formula is an generalization of the Narayana numbers.Comment: 19 pages, 12 figure
Tableaux combinatorics for two-species PASEP probabilities
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 âheavyâ and one âlightâ, 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 . 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 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 case, using certain two-species alternative tableaux.Le but de ce document est de fournir une expression combinatoire dĂ©crivant les probabilitĂ©s de lâĂ©tat dâĂ©quilibre de PASEP Ă deux espĂšces. Dans ce modĂšle, il existe deux espĂšces de particules, une âlourdeâ et une âlĂ©gĂšreâ, 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 . Par ailleurs, lorsque les particules lourdes et lĂ©gĂšres sont Ă cĂŽtĂ© lâune de lâautre, 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 , en termes de certains tableaux âmulti-Catalanâ. 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 , en utilisant certains tableaux alternatifs de deux espĂšces
Free fermionic probability theory and K-theoretic Schubert calculus
For each of the four particle processes given by Dieker and Warren
[arXiv:0707.1843], we show the -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
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
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
We study mixing times for the totally asymmetric simple exclusion process
(TASEP) on a segment of size with open boundaries. We focus on the maximal
current phase, and prove that the mixing time is of order , 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 . 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