An involution on Dyck paths and its consequences

AbstractAn involution is introduced in the set of all Dyck paths of semilength n from which one re-obtains easily the equidistribution of the parameters ‘number of valleys’ and ‘number of doublerises’ and also the equidistribution of the parameters ‘height of the first peak’ and ‘number of returns’

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 Rényi entropy, which is a measure of the partitioning of the state space, and a deformation of the familiar Shannon entropy. The Rényi 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 Rényi 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 Rényi 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 Rényi 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

Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions

We find Stieltjes-type and Jacobi-type continued fractions for some "master polynomials" that enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) number of simultaneous statistics. Our results contain many previously obtained identities as special cases, providing a common refinement of all of them.Comment: LaTeX2e, 122 pages, includes 9 tikz figure

Façonnement de l'Interférence en vue d'une Optimisation Globale d'un Système Moderne de Communication

A communication is impulsive whenever the information-bearing signal is burst-like in time. Examples of the impulsive concept are: impulse-radio signals, that is, wireless signals occurring within short intervals of time; optical signals conveyed by photons; speech signals represented by sound pressure variations; pulse-position modulated electrical signals; a sequence of arrival/departure events in a queue; neural spike trains in the brain. Understanding impulsive communications requires to identify what is peculiar to this transmission paradigm, that is, different from traditional continuous communications.In order to address the problem of understanding impulsive vs. non-impulsive communications, the framework of investigation must include the following aspects: the different interference statistics directly following from the impulsive signal structure; the different interaction of the impulsive signal with the physical medium; the actual possibility for impulsive communications of coding information into the time structure, relaxing the implicit assumption made in continuous transmissions that time is a mere support. This thesis partially addresses a few of the above issues, and draws future lines of investigation. In particular, we studied: multiple access channels where each user adopts time-hopping spread-spectrum; systems using a specific prefilter at the transmitter side, namely the transmit matched filter (also known as time reversal), particularly suited for ultrawide bandwidhts; the distribution function of interference for impulsive systems in several different settings.Une communication est impulsive chaque fois que le signal portant des informations est intermittent dans le temps et que la transmission se produit à rafales. Des exemples du concept impulsife sont : les signaux radio impulsifs, c’est-à-dire des signaux très courts dans le temps; les signaux optiques utilisé dans les systèmes de télécommunications; certains signaux acoustiques et, en particulier, les impulsions produites par le système glottale; les signaux électriques modulés en position d’impulsions; une séquence d’événements dans une file d’attente; les trains de potentiels neuronaux dans le système neuronal. Ce paradigme de transmission est différent des communications continues traditionnelles et la compréhension des communications impulsives est donc essentielle. Afin d’affronter le problème des communications impulsives, le cadre de la recherche doit inclure les aspects suivants : la statistique d’interférence qui suit directement la structure des signaux impulsifs; l’interaction du signal impulsif avec le milieu physique; la possibilité pour les communications impulsives de coder l’information dans la structure temporelle. Cette thèse adresse une partie des questions précédentes et trace des lignes indicatives pour de futures recherches. En particulier, nous avons étudié: un système d'accès multiple où les utilisateurs adoptent des signaux avec étalement de spectre par saut temporel (time-hopping spread spectrum) pour communiquer vers un récepteur commun; un système avec un préfiltre à l'émetteur, et plus précisément un transmit matched filter, également connu comme time reversal dans la littérature de systèmes à bande ultra large; un modèle d'interférence pour des signaux impulsifs