arXiv subjects

Amol Aggarwal

Publications and source records attributed to Amol Aggarwal.

45 records · Page 3Linked to original sources

arXiv2018

Current Fluctuations of the Stationary ASEP and Six-Vertex Model

Our results in this paper are two-fold. First, we consider current fluctuations of the stationary asymmetric simple exclusion process (ASEP), run for some long time $T$, and show that they are of order $T^{1 / 3}$ along a characteristic line. Upon scaling by $T^{1 / 3}$, we establish that these fluctuations converge to the long-time height fluctuations of the stationary KPZ equation, that is, to the Baik-Rains distribution. This result has long been predicted under the context of KPZ universality and in particular extends upon a number of results in the field, including the work of Ferrari and Spohn in 2005 (who established the same result for the TASEP), and the work of Balazs and Seppalainen in 2010 (who established the $T^{1 / 3}$ scaling for the general ASEP). Second, we introduce a class of translation-invariant Gibbs measures that characterizes a one-parameter family of slopes for an arbitrary ferroelectric, symmetric six-vertex model. This family of slopes corresponds to what is known as the conical singularity (or tricritical point) in the free energy profile for the ferroelectric six-vertex model. We consider fluctuations of the height function of this model on a large grid of size $T$ and show that they too are of order $T^{1 / 3}$ along a certain characteristic line; this confirms a prediction of Bukman and Shore from 1995 stating that the ferroelectric six-vertex model should exhibit KPZ growth at the conical singularity. Upon scaling the height fluctuations by $T^{1 / 3}$, we again recover the Baik-Rains distribution in the large $T$ limit. Recasting this statement in terms of the (asymmetric) stochastic six-vertex model confirms a prediction of Gwa and Spohn from 1992.

math.PR↗

arXiv2017

Dynamical Stochastic Higher Spin Vertex Models

We introduce a new family of integrable stochastic processes, called \textit{dynamical stochastic higher spin vertex models}, arising from fused representations of Felder's elliptic quantum group $E_{τ, η} (\mathfrak{sl}_2)$. These models simultaneously generalize the stochastic higher spin vertex models, studied by Corwin-Petrov and Borodin-Petrov, and are dynamical in the sense of Borodin's recent stochastic interaction round-a-face models. We provide explicit contour integral identities for observables of these models (when run under specific types of initial data) that characterize the distributions of their currents. Through asymptotic analysis of these identities in a special case, we evaluate the scaling limit for the current of a dynamical version of a discrete-time partial exclusion process. In particular, we show that its scaling exponent is $1 / 4$ and that its one-point marginal converges (in a sense of moments) to that of a non-trivial random variable, which we determine explicitly.

math-ph↗

arXiv2016

Bulk Universality for Generalized Wigner Matrices With Few Moments

In this paper we consider $N \times N$ real generalized Wigner matrices whose entries are only assumed to have finite $(2 + \varepsilon)$-th moment for some fixed, but arbitrarily small, $\varepsilon > 0$. We show that the Stieltjes transforms $m_N (z)$ of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale, when $η= \Im (z)$ is almost of order $N^{-1}$. As a consequence, we establish bulk universality for local spectral statistics of these matrices at fixed energy levels, both in terms of eigenvalue gap distributions and correlation functions, meaning that these statistics converge to those of the Gaussian Orthogonal Ensemble (GOE) in the large $N$ limit.

math.PR↗

arXiv2016

Convergence of the Stochastic Six-Vertex Model to the ASEP

In this note we establish the convergence of the stochastic six-vertex model to the one-dimensional asymmetric simple exclusion process, under a certain limit regime recently predicted by Borodin-Corwin-Gorin. This convergence holds for arbitrary initial data.

math.PR↗

arXiv2016

Phase Transitions in the ASEP and Stochastic Six-Vertex Model

In this paper we consider two models in the Kardar-Parisi-Zhang (KPZ) universality class, the asymmetric simple exclusion process (ASEP) and the stochastic six-vertex model. We introduce a new class of initial data (which we call {\itshape generalized step Bernoulli initial data}) for both of these models that generalizes the step Bernoulli initial data studied in a number of recent works on the ASEP. Under this class of initial data, we analyze the current fluctuations of both the ASEP and stochastic six-vertex model and establish the existence of a phase transition along a characteristic line, across which the fluctuation exponent changes from $1 / 2$ to $1 / 3$. On the characteristic line, the current fluctuations converge to the general (rank $k$) Baik-Ben-Arous-Péché distribution for the law of the largest eigenvalue of a critically spiked covariance matrix. For $k = 1$, this was established for the ASEP by Tracy and Widom; for $k > 1$ (and also $k = 1$, for the stochastic six-vertex model), the appearance of these distributions in both models is new.

math.PR↗

arXiv2015

Armstrong's Conjecture for $(k, mk + 1)$-Core Partitions

A conjecture of Armstrong states that if $\gcd (a, b) = 1$, then the average size of an $(a, b)$-core partition is $(a - 1)(b - 1)(a + b + 1) / 24$. Recently, Stanley and Zanello used a recursive argument to verify this conjecture when $a = b - 1$. In this paper we use a variant of their method to establish Armstrong's conjecture in the more general setting where $a$ divides $b - 1$.

math.CO↗

arXiv2014

When Does the Set of $(a, b, c)$-Core Partitions Have a Unique Maximal Element?

In 2007, Olsson and Stanton gave an explicit form for the largest $(a, b)$-core partition, for any relatively prime positive integers $a$ and $b$, and asked whether there exists an $(a, b)$-core that contains all other $(a, b)$-cores as subpartitions; this question was answered in the affirmative first by Vandehey and later by Fayers independently. In this paper we investigate a generalization of this question, which was originally posed by Fayers: for what triples of positive integers $(a, b, c)$ does there exist an $(a, b, c)$-core that contains all other $(a, b, c)$-cores as subpartitions? We completely answer this question when $a$, $b$, and $c$ are pairwise relatively prime; we then use this to generalize the result of Olsson and Stanton.

math.CO↗

arXiv2014

On Unit Distances in a Convex Polygon

In 1959, Erdős and Moser asked for the maximum number of unit distances that may be formed among the vertices of a convex $n$-gon; until now, the best known upper bound has been $2πn \log_2 n + O(n)$, achieved by Füredi in 1990. In this paper, we examine two properties that any convex polygon must satisfy and use them to prove several new facts related to the question posed by Erdős and Moser. In particular, we improve on Füredi's result, and instead obtain a bound of $n \log_2 n + O(n)$; we exhibit a class of `cycles' formed by unit distances that are forbidden in convex polygons; and we provide a lower bound that shows the limitations of our methods. The second result addresses a question asked by Fishburn and Reeds regarding the possible configurations of vertices that form a convex polygon.

cs.CG↗

arXiv2010

On Isosceles Triangles and Related Problems in a Convex Polygon

Given any convex $n$-gon, in this article, we: (i) prove that its vertices can form at most $n^2/2 + Θ(n\log n)$ isosceles trianges with two sides of unit length and show that this bound is optimal in the first order, (ii) conjecture that its vertices can form at most $3n^2/4 + o(n^2)$ isosceles triangles and prove this conjecture for a special group of convex $n$-gons, (iii) prove that its vertices can form at most $\lfloor n/k \rfloor$ regular $k$-gons for any integer $k\ge 4$ and that this bound is optimal, and (iv) provide a short proof that the sum of all the distances between its vertices is at least $(n-1)/2$ and at most $\lfloor n/2 \rfloor \lceil n/2 \rceil(1/2)$ as long as the convex $n$-gon has unit perimeter.

cs.CG↗