arXiv subjects

Arvind Ayyer

Publications and source records attributed to Arvind Ayyer.

At least 19 recordsLinked to original sources

arXiv2026

Positive definite, positive semidefinite and totally positive matrices over finite fields

Motivated by the equivalent definitions of positive definite (resp. positive semidefinite) matrices over real and complex fields, we give four (resp. five) inequivalent definitions for these matrices over finite fields. Our starting point is the recent definition due to Cooper--Hanna--Whitlatch (RMJ. Math., 2024) of positive elements in finite fields. We also use this definition to study totally positive matrices over finite fields. For all of these cases, we give explicit enumeration formulae or give bounds. Most of our formulas are new, but we summarize results from the existing literature for completeness. For positive semidefinite matrices of type 5 and totally positive matrices, we give structural formulas using the rationality of the Weil zeta function, i.e. Dwork's theorem, and conjecture a quasipolynomial-type formula.

math.RA↗

arXiv2026

A bijection on balanced words reversing both $\text{des}$ and $\text{maj}$

Balanced words on a finite alphabet are those words in which every letter of the alphabet occurs the same number of times. The notion of descents and major index extends in a natural way to words. It is known that the bivariate generating polynomials for descents and major index over balanced words on the alphabet $[k]$ with $n$ occurrences each is palindromic, but a bijective proof has been missing even for balanced binary words. We give an explicit bijection proving this result. For permutations (which are also balanced), our bijection is different from the complementation map. We also show that for balanced binary words, this bijection simultaneously flips the ascent and comajor index as well.

math.CO↗

arXiv2026

An exactly solvable evaporation-deposition PCA with long-distance interactions

We consider a probabilistic cellular automaton (PCA) of evaporation-deposition on the one-dimensional lattice having $n$ sites with periodic boundary conditions, in which each site, during each epoch, can be in one of two states: $0$ and $1$. Fix a positive integer $m\geqslant 2$. There are two types of transitions at each discrete time, which are as follows: (i) the first site in every contiguous block of $m$ $0$s becomes a $1$ with probability $p_1$, and (ii) the first site in every contiguous block of $(m-1)$ $0$s followed immediately by a $1$ also becomes a $1$ with probability $(1-p_2)$. As in a PCA, all of these transitions occur simultaneously. We show that the resulting discrete-time Markov chain is ergodic, and we give an explicit formula for its limiting distribution, the partition function and the density. We also propose necessary and sufficient conditions for this Markov chain to be reversible. For $m=2$, we provide a fully analytical expression for the free energy of this model.

math.PR↗

arXiv2026

Multispecies inhomogeneous $t$-PushTASEP with general capacity

We study an $n$-species $t$-PushTASEP, an integrable long-range stochastic process, on a one-dimensional periodic lattice with inhomogeneities $x_1,\ldots,x_L$ and arbitrary capacity $l$ at each lattice site. The Markov matrix is identified with an alternating sum of commuting transfer matrices over all fundamental representations of $U_t(\widehat{sl}_{n+1})$. Stationary probabilities are expressed in a matrix product form involving a fusion of quantized corner transfer matrices for the strange five-vertex model introduced by Okado, Scrimshaw, and the second author. The resulting partition function, which serves as the normalization factor of the stationary probabilities, is obtained from the $l=1$ case by a finite plethystic substitution of length $l$.

math-ph↗

arXiv2026

Factorised stationary states for a long range misanthrope process

The misanthrope process is an interacting particle system where particles move between neighbouring sites with hop rates depending only on the number of particles at the departure and arrival sites. Motivated by a discretised version of the Hammersley--Aldous--Diaconis process, we introduce a partially asymmetric long range misanthrope process (PALRMP) on a finite one-dimensional lattice with periodic boundary conditions where particles can move between sites that are not necessarily neighbours, as long as there are no particles in between the departure and arrival sites. In this model, each site $\ell$ has an inhomogeneous rate parameter $x_\ell$ associated to it, and the hop rate of a particle moving from site $k$ to site $\ell$ depends upon the parameter associated to the target site $x_\ell$, the direction the particle moves, and the number of particles at sites $k$ and $\ell$. We also consider the homogeneous PALRMP, where all the $x_\ell$'s are 1. We find necessary and sufficient conditions on the hop rates under which the stationary distribution is of factorised form for both the PALRMP and the homogeneous PALRMP, as well as the extreme variants, namely the ones where the particle motion is totally asymmetric (TALRMP) and symmetric (SLRMP). As an illustrative example, we study in detail the discrete Hammersley--Aldous--Diaconis process.

math.PR↗

arXiv2026

$q$-deformations of the Tsetlin library

The Tsetlin library is a random shuffling process on permutations of $n$ letters, where each letter $i$ can be interpreted as a book; book $i$ is brought to the front of the bookshelf with an assigned probability $x_i$. We define a $q$-deformation of the Tsetlin library by replacing the symmetric group action on permutations by the action of the type $A$ Iwahori-Hecke algebra. We compute the stationary distribution and spectrum of this Markov chain by relating it to a Markov chain on complete flags over the finite field vector space $\mathbb{F}_q^n$ and applying techniques from semigroup theory. We prove that for a natural choice of $x_i$ the total variation distance mixing time of the $q$-Tsetlin library on permutations of $n$ is $O(n)$ compared to $\Theta(n \log n)$ for the Tsetlin library at $q=1$, which demonstrates a phase transition. We also generalize the $q$-Tsetlin library to words (with repeated letters), and compute its stationary distribution and spectrum.

math.CO↗

arXiv2025

Dimension statistics of representations of finite groups

The first part of this paper deals with unipotent and reductive groups over finite fields with $q$ elements in which either $q$ goes to infinity or $G=GL_n(q)$ and $n$ goes to infinity. The second part of the paper deals with the symmetric group $S_n$. The main conclusion that we want to bring out in the case of reductive groups $G(q)$, $q$ varying, is that the dimension data, resp. the size of conjugacy classes, is in a statistical sense, ``roughly'' constant and the same (up to taking the squares). We introduce the notion of {\it asympototically constant}, and {\it asympototically log constant} to make precise these notions, which we apply to various groups discussed in this paper including the symmetric groups $S_n$.

math.RT↗

arXiv2025

An exactly solvable asymmetric simple inclusion process

We study a generalization of the asymmetric simple inclusion process (ASIP) on a periodic one-dimensional lattice, where the integers in the particles rates are deformed to their $t$-analogues. We call this the $(q, t, \theta)$~ASIP, where $q$ is the asymmetric hopping parameter and $\theta$ is the diffusion parameter. We show that this process is a misanthrope process, and consequently the steady state is independent of $q$. We compute the steady state, the one-point correlation and the current in the steady state. In particular, we show that the single-site occupation probabilities follow a \emph{beta-binomial} distribution at $t=1$. We compute the two-dimensional phase diagram in various regimes of the parameters $(t, \theta)$ and perform simulations to justify the results. We also show that a modified form of the steady state weights at $t \neq 1$ satisfy curious palindromic and antipalindromic symmetries. Lastly, we define an enriched process at $t=1$ and $\theta$ an integer which projects onto the $(q, 1, \theta)$~ASIP and whose steady state is uniform, which may be of independent interest.

cond-mat.stat-mech↗

arXiv2025

The bunkbed problem and the random cluster model

The well known bunkbed conjecture about percolation on finite graphs is now resolved; Gladkov, Pak and Zimin, building upon work of Hollom, have constructed a counterexample. We revisit this conjecture and study it in the broader context of the class of random cluster measures. We show that the major partial (positive) results on the bunkbed conjecture can also be proved for all random cluster measures, including the results for complete graphs, complete bipartite graphs, and the case when $p \uparrow 1$. The arboreal gas measure for forests is another limit of the random cluster measure for which we conjecture the inequality to be true and provide proofs in special cases. We identify a setting where the conjecture does hold, that of ``almost spanning tree measures''. A further analysis leads to intriguing correlation inequalities that complement Rayleigh's inequalities for spanning tree measures.

math.PR↗

arXiv2025

Multispecies inhomogeneous $t$-PushTASEP from antisymmetric fusion

We investigate the recently introduced inhomogeneous $n$-species $t$-PushTASEP, a long-range stochastic process on a periodic lattice. A Baxter-type formula is established, expressing the Markov matrix as an alternating sum of commuting transfer matrices over all the fundamental representations of $U_t(\widehat{sl}_{n+1})$. This superposition acts as an inclusion-exclusion principle, selectively extracting the sequential particle transitions characteristic of the PushTASEP, while canceling forbidden channels. The homogeneous specialization connects the PushTASEP to ASEP, showing that the two models share eigenstates and a common integrability structure.

math-ph↗

arXiv2024

Further results for classical and universal characters twisted by roots of unity

We revisit factorizations of classical characters under various specializations, some old and some new. We first show that all characters of classical families of groups twisted by odd powers of an even primitive root of unity factorize into products of characters of smaller groups. Motivated by conjectures of Wagh and Prasad (Manuscr. Math. 2020), we then observe that certain specializations of Schur polynomials factor into products of two characters of other groups. We next show, via a detour through hook Schur polynomials, that certain Schur polynomials indexed by staircase shapes factorize into linear pieces. Lastly, we consider classical and universal characters specialized at roots of unity. One of our results, in parallel with Schur polynomials, is that universal characters take values only in $\{0, \pm 1, \pm 2\}$ at roots of unity.

math.CO↗

arXiv2024

How large is the character degree sum compared to the character table sum for a finite group?

In 1961, Solomon gave upper and lower bounds for the sum of all the entries in the character table of a finite group in terms of elementary properties of the group. In a different direction, we consider the ratio of the character table sum to the sum of the entries in the first column, also known as the character degree sum, in this work. First, we propose that this ratio is at most two for many natural groups. Secondly, we extend a conjecture of Fields to postulate that this ratio is at least one with equality if and only if the group is abelian. We establish the validity of this property and conjecture for all finite irreducible Coxeter groups. In addition, we prove the conjecture for generalized symmetric groups. The main tool we use is that the sum of a column in the character table of an irreducible Coxeter group (resp. generalized symmetric group) is given by the number of square roots (resp. absolute square roots) of the corresponding conjugacy class representative. As a byproduct of our results, we show that the asymptotics of character table sums is the same as the number of involutions in symmetric, hyperoctahedral and demihyperoctahedral groups. We also derive explicit generating functions for the character table sums for these latter groups as infinite products of continued fractions. In the same spirit, we prove similar generating function formulas for the number of square roots and absolute square roots in $n$ for the generalized symmetric groups $G(r,1,n)$.

math.RT↗

arXiv2024

The inhomogeneous $t$-PushTASEP and Macdonald polynomials

We study a multispecies $t$-PushTASEP system on a finite ring of $n$ sites with site-dependent rates $x_1,\dots,x_n$. Let $\lambda=(\lambda_1,\dots,\lambda_n)$ be a partition whose parts represent the species of the $n$ particles on the ring. We show that for each composition $\eta$ obtained by permuting the parts of $\lambda$, the stationary probability of being in state $\eta$ is proportional to the ASEP polynomial $F_{\eta}(x_1,\dots,x_n; q,t)$ at $q=1$; the normalizing constant (or partition function) is the Macdonald polynomial $P_{\lambda}(x_1,\dots,x_n;q,t)$ at $q=1$. Our approach involves new relations between the families of ASEP polynomials and of non-symmetric Macdonald polynomials at $q=1$. We also use multiline diagrams, showing that a single jump of the PushTASEP system is closely related to the operation of moving from one line to the next in a multiline diagram. We derive symmetry properties for the system under permutation of its jump rates, as well as a formula for the current of a single-species system.

math.CO↗

arXiv2024

An area-bounce exchanging bijection on a large subset of Dyck paths

It is a longstanding open problem to find a bijection exchanging area and bounce statistics on Dyck paths. We settle this problem for an exponentially large subset of Dyck paths via an explicit bijection. Moreover, we prove that this bijection is natural by showing that it maps what we call bounce-minimal paths to area-minimal paths. As a consequence of the proof ideas, we show combinatorially that a path with area $a$ and bounce $b$ exists if and only if a path with area $b$ and bounce $a$ exists. We finally show that the number of distinct values of the sum of the area and bounce statistics is the number of nonzero coefficients in Johnson's $q$-Bell polynomial.

math.CO↗

arXiv2023

The inhomogeneous multispecies PushTASEP: Dynamics and symmetry

We introduce and study a natural multispecies variant of the inhomogeneous PushTASEP with site-dependent rates on the finite ring. We show that the stationary distribution of this process is proportional to the ASEP polynomials at $q = 1$ and $t = 0$. This is done by constructing a multiline process which projects to the multispecies PushTASEP, and identifying its stationary distribution using time-reversal arguments. We also study symmetry properties of the process under interchange of the rates associated to the sites. These results hold not just for events depending on the configuration at a single time in equilibrium, but also for systems out of equilibrium and for events depending on the path of the process over time. Lastly, we give explicit formulas for nearest-neighbour two-point correlations in terms of Schur functions.

math.PR↗

arXiv2023

An exactly solvable asymmetric $K$-exclusion process

We study an interacting particle process on a finite ring with $L$ sites with at most $K$ particles per site, in which particles hop to nearest neighbors with rates given in terms of $t$-deformed integers and asymmetry parameter $q$, where $t>0$ and $q \geq 0$ are parameters. This model, which we call the $(q, t)$~$K$-ASEP, reduces to the usual ASEP on the ring when $K = 1$ and to a model studied by Sch\"utz and Sandow (\emph{Phys. Rev. E}, 1994) when $t = q = 1$. This is a special case of the misanthrope process and as a consequence, the steady state does not depend on $q$ and is of product form, generalizing the same phenomena for the ASEP. What is interesting here is the steady state weights are given by explicit formulas involving $t$-binomial coefficients, and are palindromic polynomials in $t$. Interestingly, although the $(q, t)$~$K$-ASEP does not satisfy particle-hole symmetry, its steady state does. We analyze the density and calculate the most probable number of particles at a site in the steady state in various regimes of $t$. Lastly, we construct a two-dimensional exclusion process on a discrete cylinder with height $K$ and circumference $L$ which projects to the $(q, t)$~$K$-ASEP and whose steady state distribution is also of product form. We believe this model will serve as an illustrative example in constructing two-dimensional analogues of misanthrope processes. Simulations are attached as ancillary files.

cond-mat.stat-mech↗

arXiv2023

Combinatorial proofs of multivariate Cayley--Hamilton theorems

We give combinatorial proofs of two multivariate Cayley--Hamilton type theorems. The first one is due to Phillips (Amer. J. Math., 1919) involving $2k$ matrices, of which $k$ commute pairwise. The second one regards the mixed discriminant, a matrix function which has generated a lot of interest in recent times. Recently, the Cayley--Hamilton theorem for mixed discriminants was proved by Bapat and Roy (Comb. Math. and Comb. Comp., 2017). We prove a Phillips-type generalization of the Bapat--Roy theorem involving $2nk$ matrices, where $n$ is the size of the matrices, among which $nk$ commute pairwise. Our proofs generalize the univariate proof of Straubing (Disc. Math., 1983) for the original Cayley--Hamilton theorem in a nontrivial way, and involve decorated permutations and decorated paths.

math.CO↗

arXiv2022

Cores of partitions in rectangles

For a positive integer $t \geq 2$, the $t$-core of a partition plays an important role in modular representation theory and combinatorics. We initiate the study of $t$-cores of partitions contained in an $r \times s$ rectangle. Our main results are as follows. We first give a simple formula for the number of partitions in the rectangle that are themselves $t$-cores and compute its asymptotics for large $r,s$. We then prove that the number of partitions inside the rectangle whose $t$-cores are a fixed partition $\rho$ is given by a product of binomial coefficients. Finally, we use this formula to compute the distribution of the $t$-core of a uniformly random partition inside the rectangle extending our previous work on all partitions of a fixed integer $n$ (Ann. Appl. Prob. 2023). In particular, we show that in the limit as $r,s \to \infty$ maintaining a fixed aspect ratio, we again obtain a Gamma distribution with the same shape parameter $\alpha = (t-1)/2$ and rate parameter $\beta$ that depends on the aspect ratio.

math.CO↗