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Harold Widom

Publications and source records attributed to Harold Widom.

At least 19 recordsLinked to original sources
arXiv2022

Domain Walls in the Heisenberg-Ising Spin-1/2 Chain

In this paper we obtain formulas for the distribution of the left-most up-spin in the Heisenberg-Ising spin-1/2 chain with anisotropy parameter $Δ$, also known as the XXZ spin-1/2 chain, on the one-dimensional lattice $\mathbb{Z}$ with domain wall initial conditions. We use the Bethe Ansatz to solve the Schr$ö$dinger equation and a recent antisymmetrization identity of Cantini, Colomo, and Pronko (arXiv:1906.07636) to simplify the marginal distribution of the left-most up-spin. In the $Δ=0$ case, the distribution $F_2$ arises. In the $Δ\neq 0$ case, we propose a conjectural series expansion type formula based on a saddle point analysis. The conjectural formula turns out to be a Fredholm series expansion in the $Δ\rightarrow 0$ limit and recovers the result for $Δ= 0$.

math.PR↗
arXiv2017

Blocks and Gaps in the Asymmetric Simple Exclusion Process: Asymptotics

In earlier work (arXiv:1707.04927) the authors obtained formulas for the probability in the asymmetric simple exclusion process that at time $t$ a particle is at site $x$ and is the beginning of a block of $L$ consecutive particles. Here we consider asymptotics. Specifically, for the KPZ regime with step initial condition, we determine the conditional probability (asymptotically as $t\rightarrow\infty$) that a particle is the beginning of an $L$-block, given that it is at site $x$ at time $t$. Using duality between occupied and unoccupied sites we obtain the analogous result for a gap of $G$ unoccupied sites between the particle at $x$ and the next one.

math-ph↗
arXiv2017

Blocks in the Asymmetric Simple Exclusion Process

In earlier work the authors obtained formulas for the probability in the asymmetric simple exclusion process that the $m$th particle from the left is at site $x$ at time $t$. They were expressed in general as sums of multiple integrals and, for the case of step initial condition, as an integral involving a Fredholm determinant. In the present work these results are generalized to the case where the $m$th particle is the left-most one in a contiguous block of $L$ particles. The earlier work depended in a crucial way on two combinatorial identities, and the present work begins with a generalization of these identities to general $L$.

math-ph↗
arXiv2016

On the ground state energy of the delta-function Fermi gas

The weak coupling asymptotics to order $γ$ of the ground state energy of the delta-function Fermi gas, derived heuristically in the literature, is here made rigorous. Further asymptotics are in principle computable. The analysis applies to the Gaudin integral equation a method previously used by one of the authors for the asymptotics of large Toeplitz matrices.

math-ph↗
arXiv2016

On the ground state energy of the delta-function Bose gas

The weak coupling asymptotics, to order $(c/ρ)^2$, of the ground state energy of the delta-function Bose gasmis derived. Here $2c\ge 0$ is the delta-function potential amplitude and $ρ$ the density of the gas in the thermodynamic limit. The analysis uses the electrostatic interpretation of the Lieb-Liniger integral equation.

math-ph↗
arXiv2015

Useful bounds on the extreme eigenvalues and vectors of matrices for Harper's operators

In analyzing a simple random walk on the Heisenberg group we encounter the problem of bounding the extreme eigenvalues of an $n\times n$ matrix of the form $M=C+D$ where $C$ is a circulant and $D$ a diagonal matrix. The discrete Schrödinger operators are an interesting special case. The Weyl and Horn bounds are not useful here. This paper develops three different approaches to getting good bounds. The first uses the geometry of the eigenspaces of $C$ and $D$, applying a discrete version of the uncertainty principle. The second shows that, in a useful limit, the matrix $M$ tends to the harmonic oscillator on $L^2(\mathbb{R})$ and the known eigenstructure can be transferred back. The third approach is purely probabilistic, extending $M$ to an absorbing Markov chain and using hitting time arguments to bound the Dirichlet eigenvalues. The approaches allow generalization to other walks on other groups.

math.PR↗
arXiv2015

Natural Boundary for a Sum Involving Toeplitz Determinants

In the theory of the two-dimensional Ising model, the diagonal susceptibility is equal to a sum involving Toeplitz determinants. In terms of a parameter k the diagonal susceptibility is analytic inside the unit circle, and the authors proved the conjecture that this function has the unit circle as a natural boundary. The symbol of the Toepltiz determinants was a k-deformation of one with a single singularity on the unit circle. Here we extend the result, first, to deformations of a larger class of symbols with a single singularity on the unit circle, and then to deformations of (almost) general Fisher-Hartwig symbols.

math-ph↗
arXiv2015

An Exercise (?) in Fourier Analysis on the Heisenberg Group

Let H(n) be the group of 3x3 uni-uppertriangular matrices with entries in Z/nZ, the integers mod n. We show that the simple random walk converges to the uniform distribution in order n^2 steps. The argument uses Fourier analysis and is surprisingly challenging. It introduces novel techniques for bounding the spectrum which are useful for a variety of walks on a variety of groups.

math.PR↗
arXiv2014

On the Singularities in the Susceptibility Expansion for the Two-Dimensional Ising Model

For temperatures below the critical temperature, the magnetic susceptibility for the two-dimensional isotropic Ising model can be expressed in terms of an infinite series of multiple integrals. With respect to a parameter related to temperature and the interaction constant, the integrals may be extended to functions analytic outside the unit circle. In a groundbreaking paper, B. G. Nickel identified a class of singularities of these integrals on the unit circle. In this note we show that there are no other singularities on the unit circle.

math-ph↗
arXiv2013

On the diagonal susceptibility of the 2D Ising model

We consider the diagonal susceptibility of the isotropic 2D Ising model for temperatures below the critical temperature. For a parameter k related to temperature and the interaction constant, we extend the diagonal susceptibility to complex k inside the unit disc, and prove the conjecture that the unit circle is a natural boundary.

math-ph↗
arXiv2013

The asymmetric simple exclusion process with an open boundary

We consider the asymmetric simple exclusion process confined to the nonnegative integers with an open boundary at 0. The point 0 is connected to a reservoir where particles are injected and ejected at prescribed rates subject to the exclusion rule. We derive formulas for the transition probability as a function of time from states where initially there are m particles to states where there are n particles.

math.PR↗
arXiv2012

On the asymmetric simple exclusion process with multiple species

In the asymmetric simple exclusion process on the integers each particle waits exponential time, then with probability p it moves one step to the right if the site is unoccupied, otherwise it stays put; and with probability q=1-p it moves one step to the left if the site is unoccupied, otherwise it stays put. In previous work the authors, using the Bethe Ansatz, found for N-particle ASEP a formula --- a sum of multiple integrals --- for the probability that a system is in a particular configuration at time t given an initial configuration. The present work extends this to the case where particles are of different species, with particles of a higher species having priority over those of a lower species. Here the integrands in the multiple integrals are defined by a system of relations whose consistency requires verifying that the Yang-Baxter equations are satisfied.

math.PR↗
arXiv2012

The Bose Gas and Asymmetric Simple Exclusion Process on the Half-Line

In this paper we find explicit formulas for: (1) Green's function for a system of one-dimensional bosons interacting via a delta-function potential with particles confined to the positive half-line; and (2) the transition probability for the one-dimensional asymmetric simple exclusion process (ASEP) with particles confined to the nonnegative integers. These are both for systems with a finite number of particles. The formulas are analogous to ones obtained earlier for the Bose gas and ASEP on the line and integers, respectively. We use coordinate Bethe Ansatz appropriately modified to account for confinement of the particles to the half-line. As in the earlier work, the proof for the ASEP is less straightforward than for the Bose gas.

math.PR↗
arXiv2011

Formulas and Asymptotics for the Asymmetric Simple Exclusion Process

This is an expanded version of a series of lectures delivered by the second author in June, 2009. It describes the results of three of the authors' papers on ASEP, from the derivation of exact formulas for configuration probabilities, through Fredholm determinant representation, to asymptotics for ASEP with step initial condition establishing KPZ universality. Although complete proofs are in general not given, at least the main elements of them are.

math.PR↗
arXiv2011

Integral Formulas for the Asymmetric Simple Exclusion Process

In this paper we obtain general integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice with nearest neighbor hopping rates p to the right and q=1-p to the left. For the most part we consider an N-particle system but for certain of these formulas we can take the limit as N goes to infinity. First we obtain, for the N-particle system, a formula for the probability of a configuration at time t, given the initial configuration. For this we use Bethe Ansatz ideas to solve the master equation, extending a result of Schuetz for the case N=2. The main results of the paper, derived from this, are integral formulas for the probability, for given initial configuration, that the m'th left-most particle is at x at time t. In one of these formulas we can take the limit as N goes to infinity, and it gives the probability for an infinite system where the initial configuration is bounded on one side. For the special case of the totally asymmetric simple exclusion process (TASEP) our formulas reduce to the known ones.

math.PR↗
arXiv2010

On ASEP with Periodic Step Bernoulli Initial Condition

We consider the asymmetric simple exclusion process (ASEP) on the integers in which the initial density at a site (the probability that it is occupied) is given by a periodic function on the positive integers. (When the function is constant this is the step Bernoulli initial condition.) Starting with a result in earlier work we find a formula for the probability distribution for a given particle at a given time which is a sum over positive integers k of integrals of order k.

math.PR↗
arXiv2010

Formulas for Joint Probabilities for the Asymmetric Simple Exclusion Process

In earlier work the authors obtained integral formulas for probabilities for a single particle in the asymmetric simple exclusion process. Here formulas are obtained for joint probabilities for several particles. In the case of a single particle the derivation here is simpler than the one in the earlier work for one of its main results.

math.PR↗