arXiv subjects

Arvind Ayyer

Publications and source records attributed to Arvind Ayyer.

At least 55 records · Page 3Linked to original sources

arXiv2019

The exact phase diagram for a semipermeable TASEP with nonlocal boundary jumps

We consider a finite one-dimensional totally asymmetric simple exclusion process (TASEP) with four types of particles, $\{1,0,\bar{1},*\}$, in contact with reservoirs. Particles of species $0$ can neither enter nor exit the lattice, and those of species $*$ are constrained to lie at the first and last site. Particles of species $1$ enter from the left reservoir into either the first or second site, move rightwards, and leave from either the last or penultimate site. Conversely, particles of species $\bar{1}$ enter from the right reservoir into either the last or penultimate site, move leftwards, and leave from either the first or last site. This dynamics is motivated by a natural random walk on the Weyl group of type D. We compute the exact nonequilibrium steady state distribution using a matrix ansatz building on earlier work of Arita. We then give explicit formulas for the nonequilibrium partition function as well as densities and currents of all species in the steady state, and derive the phase diagram.

cond-mat.stat-mech↗

arXiv2019

Extensions of partial cyclic orders and consecutive coordinate polytopes

We introduce several classes of polytopes contained in $[0,1]^n$ and cut out by inequalities involving sums of consecutive coordinates. We show that the normalized volumes of these polytopes enumerate circular extensions of certain partial cyclic orders. Among other things this gives a new point of view on a question popularized by Stanley. We also provide a combinatorial interpretation of the Ehrhart $h^*$-polynomials of some of these polytopes in terms of descents of total cyclic orders. The Euler numbers, the Eulerian numbers and the Narayana numbers appear as special cases.

math.CO↗

arXiv2019

Random motion on finite rings, I: commutative rings

We consider irreversible Markov chains on finite commutative rings randomly generated using both addition and multiplication. We restrict ourselves to the case where the addition is uniformly random and multiplication is arbitrary. We first prove formulas for eigenvalues and multiplicities of the transition matrices of these chains using the character theory of finite abelian groups. The examples of principal ideal rings (such as $\mathbb{Z}_{n}$) and finite chain rings (such as $\mathbb{Z}_{p^k}$) are particularly illuminating and are treated separately. We then prove a recursive formula for the stationary probabilities for any ring, and use it to prove explicit formulas for the probabilities for finite chain rings when multiplication is also uniformly random. Finally, we prove constant mixing time for our chains using coupling.

math.RT↗

arXiv2019

Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions

We show that, for a certain class of partitions and an even number of variables of which half are reciprocals of the other half, Schur polynomials can be factorized into products of odd and even orthogonal characters. We also obtain related factorizations involving sums of two Schur polynomials, and certain odd-sized sets of variables. Our results generalize the factorization identities proved by Ciucu and Krattenthaler (Advances in combinatorial mathematics, 39-59, 2009) for partitions of rectangular shape. We observe that if, in some of the results, the partitions are taken to have rectangular or double-staircase shapes and all of the variables are set to 1, then factorization identities for numbers of certain plane partitions, alternating sign matrices and related combinatorial objects are obtained.

math.CO↗

arXiv2019

Random motion on finite rings, II: Noncommutative rings

We extend our previous study of Markov chains on finite commutative rings (arXiv:1605.05089) to arbitrary finite rings with identity. At each step, we either add or multiply by a randomly chosen element of the ring, where the addition (resp. multiplication) distribution is uniform (resp. conjugacy invariant). We prove explicit formulas for some of the eigenvalues of the transition matrix and give lower bounds on their multiplicities. We also give recursive formulas for the stationary distribution and prove that the mixing time is bounded by an absolute constant. For the matrix rings $M_2(\mathbb F_q),$ we compute the entire spectrum explicitly using the representation theory of $\text{GL}_2(\mathbb F_q),$ as well as the stationary probabilities.

math.RT↗

arXiv2018

Macdonald trees and determinants of representations for finite Coxeter groups

Every irreducible odd dimensional representation of the $n$'th symmetric or hyperoctahedral group, when restricted to the $(n-1)$'th, has a unique irreducible odd-dimensional constituent. Furthermore, the subgraph induced by odd-dimensional representations in the Bratteli diagram of symmetric and hyperoctahedral groups is a binary tree with a simple recursive description. We survey the description of this tree, known as the Macdonald tree, for symmetric groups, from our earlier work. We describe analogous results for hyperoctahedral groups. A partition $λ$ of $n$ is said to be chiral if the corresponding irreducible representation $V_λ$ of $S_n$ has non-trivial determinant. We review our previous results on the structure and enumeration of chiral partitions, and subsequent extension to all Coxeter groups by Ghosh and Spallone. Finally we show that the numbers of odd and chiral partitions track each other closely.

math.RT↗

arXiv2018

The phase diagram for a multispecies left-permeable asymmetric exclusion process

We study a multispecies generalization of a left-permeable asymmetric exclusion process (LPASEP) in one dimension with open boundaries. We determine all phases in the phase diagram using an exact projection to the LPASEP solved by us in a previous work. In most phases, we observe the phenomenon of dynamical expulsion of one or more species. We explain the density profiles in each phase using interacting shocks. This explanation is corroborated by simulations.

cond-mat.stat-mech↗

arXiv2018

Reverse juggling processes

Knutson introduced two families of reverse juggling Markov chains (single and multispecies) motivated by the study of random semi-infinite matrices over $\mathbb{F}_q$. We present natural generalizations of both chains by placing generic weights that still lead to simple combinatorial expressions for the stationary distribution. For permutations, this is a seemingly new multivariate generalization of the inversion polynomial.

math.PR↗

arXiv2018

The exact phase diagram for a class of multispecies asymmetric exclusion processes

The asymmetric exclusion process is an idealised stochastic model of transport, whose exact solution has given important insight into a general theory of nonequilibrium statistical physics. In this work, we consider a totally asymmetric exclusion process with multiple species of particles on a one-dimensional lattice in contact with reservoirs. We derive the exact nonequilibrium phase diagram for the system in the long time limit. We find two new phenomena in certain regions of the phase diagram: when the density of a species becomes zero throughout the system, and dynamical localisation when the density of a species is nonzero only within an interval far from the boundaries. We give a complete explanation of the macroscopic features of the phase diagram using what we call nested fat shocks.

cond-mat.stat-mech↗

arXiv2018

Bumping sequences and multispecies juggling

Building on previous work by four of us (ABCN), we consider further generalizations of Warrington's juggling Markov chains. We first introduce "multispecies" juggling, which consist in having balls of different weights: when a ball is thrown it can possibly bump into a lighter ball that is then sent to a higher position, where it can in turn bump an even lighter ball, etc. We both study the case where the number of balls of each species is conserved and the case where the juggler sends back a ball of the species of its choice. In this latter case, we actually discuss three models: add-drop, annihilation and overwriting. The first two are generalisations of models presented in (ABCN) while the third one is new and its Markov chain has the ultra fast convergence property. We finally consider the case of several jugglers exchanging balls. In all models, we give explicit product formulas for the stationary probability and closed form expressions for the normalisation factor if known.

math.CO↗

arXiv2018

Matrix product solution of a left-permeable two-species asymmetric exclusion process

We study a two-species partially asymmetric exclusion process where the left boundary is permeable for the `slower' species but the right boundary is not. We find a matrix product solution for the stationary state, and the exact stationary phase diagram for the densities and currents. By calculating the density of each species at the boundaries, we find further structure in the stationary phases. In particular, we find that the slower species can reach and accumulate at the far boundary, even in phases where the bulk density of these particles approaches zero.

cond-mat.stat-mech↗

arXiv2017

The Hilbert-Galton board

We introduce the Hilbert-Galton board as a variant of the classical Galton board. Balls fall into a row of bins at a rate depending on the bin, and at random times, each bin gets shifted one unit to the right and an empty bin is added to the left. We compute the stationary distribution of this Markov chain and show the existence of an enriched Markov chain on triangular arrays of numbers which projects down to the Hilbert-Galton board. We also define finite-ball projections of the Hilbert-Galton board, for which we compute the stationary distribution, the full spectrum and the grand coupling time.

math.PR↗

arXiv2016

Odd partitions in Young's lattice

We show that the subgraph induced in Young's graph by the set of partitions with an odd number of standard Young tableaux is a binary tree. This tree exhibits self-similarities at all scales, and has a simple recursive description.

math.CO↗

arXiv2016

Representations of symmetric groups with non-trivial determinant

We give a closed formula for the number of partitions $λ$ of $n$ such that the corresponding irreducible representation $V_λ$ of $S_n$ has non-trivial determinant. We determine how many of these partitions are self-conjugate and how many are hooks. This is achieved by characterizing the $2$-core towers of such partitions. We also obtain a formula for the number of partitions of $n$ such that the associated permutation representation of $S_n$ has non-trivial determinant.

math.RT↗

arXiv2015

Full Current Statistics for a Disordered Open Exclusion Process

We consider the nonabelian sandpile model defined on directed trees by Ayyer, Schilling, Steinberg and Thiéry (Commun. Math. Phys, 2013) and restrict it to the special case of a one-dimensional lattice of $n$ sites which has open boundaries and disordered hopping rates. We focus on the joint distribution of the integrated currents across each bond simultaneously, and calculate its cumulant generating function exactly. Surprisingly, the process conditioned on seeing specified currents across each bond turns out to be a renormalised version of the same process. We also remark on a duality property of the large deviation function. Lastly, all eigenvalues and both Perron eigenvectors of the tilted generator are determined.

cond-mat.stat-mech↗

arXiv2015

Spectral gap for random-to-random shuffling on linear extensions

In this paper, we propose a new Markov chain which generalizes random-to-random shuffling on permutations to random-to-random shuffling on linear extensions of a finite poset of size $n$. We conjecture that the second largest eigenvalue of the transition matrix is bounded above by $(1+1/n)(1-2/n)$ with equality when the poset is disconnected. This Markov chain provides a way to sample the linear extensions of the poset with a relaxation time bounded above by $n^2/(n+2)$ and a mixing time of $O(n^2 \log n)$. We conjecture that the mixing time is in fact $O(n \log n)$ as for the usual random-to-random shuffling.

math.PR↗

arXiv2015

A Statistical Model of Current Loops and Magnetic Monopoles

We formulate a natural model of current loops and magnetic monopoles for arbitrary planar graphs, which we call the monopole-dimer model, and express the partition function of this model as a determinant. We then extend the method of Kasteleyn and Temperley-Fisher to calculate the partition function exactly in the case of rectangular grids. This partition function turns out to be a square of the partition function of an emergent monomer-dimer model when the grid sizes are even. We use this formula to calculate the local monopole density, free energy and entropy exactly. Our technique is a novel determinantal formula for the partition function of a model of vertices and loops for arbitrary graphs.

cond-mat.stat-mech↗

arXiv2015

A finite variant of the Toom Model

We present results for a finite variant of the one-dimensional Toom model with closed boundaries. We show that the steady state distribution is not of product form, but is nonetheless simple. In particular, we give explicit formulas for the densities and some nearest neighbour correlation functions. We also give exact results for eigenvalues and multiplicities of the transition matrix using the theory of ${\mathscr R}$-trivial monoids in joint work with A. Schilling, B. Steinberg and N. M. Thiéry.

cond-mat.stat-mech↗