arXiv subjects

Tom Kennedy

Publications and source records attributed to Tom Kennedy.

At least 37 records · Page 2Linked to original sources

arXiv2011

The self-avoiding walk in a strip

We review the existence of the infinite length self-avoiding walk in the half plane and its relationship to bridges. We prove that this probability measure is also given by the limit as $β\rightarrow β_c-$ of the probability measure on all finite length walks $ω$ with the probability of $ω$ proportional to $β_c^{|ω|}$ where $|ω|$ is the number of steps in $ω$. The self-avoiding walk in a strip $\{z : 0<\Im(z)<y\}$ is defined by considering all self-avoiding walks $ω$ in the strip which start at the origin and end somewhere on the top boundary with probability proportional to $β_c^{|ω|}$ We prove that this probability measure may be obtained by conditioning the SAW in the half plane to have a bridge at height $y$. This observation is the basis for simulations to test conjectures on the distribution of the endpoint of the SAW in a strip and the relationship between the distribution of this strip SAW and SLE$_{8/3}$.

math.PR↗

arXiv2011

Transforming fixed-length self-avoiding walks into radial SLE_8/3

We conjecture a relationship between the scaling limit of the fixed-length ensemble of self-avoiding walks in the upper half plane and radial SLE with kappa=8/3 in this half plane from 0 to i. The relationship is that if we take a curve from the fixed-length scaling limit of the SAW, weight it by a suitable power of the distance to the endpoint of the curve and then apply the conformal map of the half plane that takes the endpoint to i, then we get the same probability measure on curves as radial SLE. In addition to a non-rigorous derivation of this conjecture, we support it with Monte Carlo simulations of the SAW. Using the conjectured relationship between the SAW and radial SLE, our simulations give estimates for both the interior and boundary scaling exponents. The values we obtain are within a few hundredths of a percent of the conjectured values.

math.PR↗

arXiv2010

Renormalization group maps for Ising models in lattice gas variables

Real space renormalization group maps, e.g., the majority rule transformation, map Ising type models to Ising type models on a coarser lattice. We show that each coefficient of the renormalized Hamiltonian in the lattice gas variables depends on only a finite number of values of the renormalized Hamiltonian. We introduce a method which computes the values of the renormalized Hamiltonian with high accuracy and so computes the coefficients in the lattice gas variables with high accuracy. For the critical nearest neighbor Ising model on the square lattice with the majority rule transformation, we compute over 1,000 different coefficients in the lattice gas variable representation of the renormalized Hamiltonian and study the decay of these coefficients. We find that they decay exponentially in some sense but with a slow decay rate. We also show that the coefficients in the spin variables are sensitive to the truncation method used to compute them.

math-ph↗

arXiv2009

Numerical computations for the Schramm-Loewner Evolution

We review two numerical methods related to the Schramm-Loewner evolution (SLE). The first simulates SLE itself. More generally, it finds the curve in the half-plane that results from the Loewner equation for a given driving function. The second method can be thought of as the inverse problem. Given a simple curve in the half-plane it computes the driving function in the Loewner equation. This algorithm can be used to test if a given random family of curves in the half-plane is SLE by computing the driving process for the curves and testing if it is Brownian motion. More generally, this algorithm can be used to compute the driving process for random curves that may not be SLE. Most of the material presented here has appeared before. Our goal is to give a pedagogic review, illustrate some of the practical issues that arise in these computations and discuss some open problems.

math-ph↗

arXiv2008

Conditioning SLEs and loop erased random walks

We discuss properties of dipolar SLE(k) under conditioning. We show that k=2, which describes continuum limits of loop erased random walks, is characterized as being the only value of k such that dipolar SLE conditioned to stop on an interval coincides with dipolar SLE on that interval. We illustrate this property by computing a new bulk passage probability for SLE(2).

math-ph↗

arXiv2008

Computing the Loewner driving process of random curves in the half plane

We simulate several models of random curves in the half plane and numerically compute their stochastic driving process (as given by the Loewner equation). Our models include models whose scaling limit is the Schramm-Loewner evolution (SLE) and models for which it is not. We study several tests of whether the driving process is Brownian motion. We find that just testing the normality of the process at a fixed time is not effective at determining if the process is Brownian motion. Tests that involve the independence of the increments of Brownian motion are much more effective. We also study the zipper algorithm for numerically computing the driving function of a simple curve. We give an implementation of this algorithm which runs in a time O(N^1.35) rather than the usual O(N^2), where N is the number of points on the curve.

math.PR↗

arXiv2007

The Length of an SLE - Monte Carlo Studies

The scaling limits of a variety of critical two-dimensional lattice models are equal to the Schramm-Loewner evolution (SLE) for a suitable value of the parameter kappa. These lattice models have a natural parametrization of their random curves given by the length of the curve. This parametrization (with suitable scaling) should provide a natural parametrization for the curves in the scaling limit. We conjecture that this parametrization is also given by a type of fractal variation along the curve, and present Monte Carlo simulations to support this conjecture. Then we show by simulations that if this fractal variation is used to parametrize the SLE, then the parametrized curves have the same distribution as the curves in the scaling limit of the lattice models with their natural parametrization.

math.PR↗

arXiv2007

A Fast Algorithm for Simulating the Chordal Schramm-Loewner Evolution

The Schramm-Loewner evolution (SLE) can be simulated by dividing the time interval into N subintervals and approximating the random conformal map of the SLE by the composition of N random, but relatively simple, conformal maps. In the usual implementation the time required to compute a single point on the SLE curve is O(N). We give an algorithm for which the time to compute a single point is O(N^p) with p<1. Simulations with kappa=8/3 and kappa=6 both give a value of p of approximately 0.4.

math.PR↗

arXiv2005

Monte Carlo comparisons of the self-avoiding walk and SLE as parameterized curves

The scaling limit of the two-dimensional self-avoiding walk (SAW) is believed to be given by the Schramm-Loewner evolution (SLE) with the parameter kappa equal to 8/3. The scaling limit of the SAW has a natural parameterization and SLE has a standard parameterization using the half-plane capacity. These two parameterizations do not correspond with one another. To make the scaling limit of the SAW and SLE agree as parameterized curves, we must reparameterize one of them. We present Monte Carlo results that show that if we reparameterize the SAW using the half-plane capacity, then it agrees well with SLE with its standard parameterization. We then consider how to reparameterize SLE to make it agree with the SAW with its natural parameterization. We argue using Monte Carlo results that the so-called p-variation of the SLE curve with p=1/nu=4/3 provides a parameterization that corresponds to the natural parameterization of the SAW.

math.PR↗

arXiv2004

A densest compact planar packing with two sizes of discs

We consider packings of the plane using discs of radius 1 and r=0.545151... . The value of r admits compact packings in which each hole in the packing is formed by three discs which are tangent to each other. We prove that the largest density possible is that of the compact packing shown in figure 1.

math.MG↗

arXiv2004

Compact packings of the plane with two sizes of discs

We consider packings of the plane using discs of radius 1 and r. A packing is compact if every disc D is tangent to a sequence of discs D_1, D_2, ..., D_n such that D_i is tangent to D_{i+1}. We prove that there are only nine values of r with r<1 for which such packings are possible. For each of the nine values we describe the possible compact packings.

math.MG↗

arXiv2003

Expansions for Droplet States in the Ferromagnetic XXZ Heisenberg Chain

We consider the highly anisotropic ferromagnetic spin 1/2 Heisenberg chain with periodic boundary conditions. In each sector of constant total z component of the spin, we develop convergent expansions for the lowest band of eigenvalues and eigenfunctions. These eigenstates describe droplet states in which the spins essentially form a single linear droplet which can move. Our results also give a convergent expansion for the dispersion relation, i.e., the energy of the droplet as a function of its momentum. The methods used are from math-ph/0208026 and cond-mat/0104199, and this short paper should serve as a pedagogic introduction to those papers.

math-ph↗

arXiv2003

Conformal Invariance and Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk - Monte Carlo Tests

Simulations of the self-avoiding walk (SAW) are performed in a half-plane and a cut-plane (the complex plane with the positive real axis removed) using the pivot algorithm. We test the conjecture of Lawler, Schramm and Werner that the scaling limit of the two-dimensional SAW is given by Schramm's Stochastic Loewner Evolution (SLE). The agreement is found to be excellent. The simulations also test the conformal invariance of the SAW since conformal invariance would imply that if we map the walks in the cut-plane into the half plane using the conformal map z -> sqrt(z), then the resulting walks will have the same distribution as the SAW in the half plane. The simulations show excellent agreement between the distributions.

math.PR↗

arXiv2003

Expansions for one quasiparticle states in spin 1/2 systems

Convergent expansions of the wavefunctions for the ground state and low-lying excited states of quantum transverse Ising systems are obtained. These expansions are employed to prove that the dispersion relation for a single quasi-particle has a convergent expansion.

cond-mat.stat-mech↗

arXiv2003

Instability of interfaces in the antiferromagnetic XXZ chain at zero temperature

For the antiferromagnetic, highly anisotropic XZ and XXZ quantum spin chains, we impose periodic boundary conditions on chains with an odd number of sites to force an interface (or kink) into the chain. We prove that the energy of the interface depends on the momentum of the state. This shows that at zero temperature the interface in such chains is not stable. This is in contrast to the ferromagnetic XXZ chain for which the existence of localized interface ground states has been proven for any amount of anisotropy in the Ising-like regime.

math-ph↗

arXiv2001

Monte Carlo Tests of SLE Predictions for the 2D Self-Avoiding Walk

The conjecture that the scaling limit of the two-dimensional self-avoiding walk (SAW) in a half plane is given by the stochastic Loewner evolution (SLE) with $κ=8/3$ leads to explicit predictions about the SAW. A remarkable feature of these predictions is that they yield not just critical exponents, but probability distributions for certain random variables associated with the self-avoiding walk. We test two of these predictions with Monte Carlo simulations and find excellent agreement, thus providing numerical support to the conjecture that the scaling limit of the SAW is SLE$_{8/3}$.

math.PR↗

arXiv2001

A faster implementation of the pivot algorithm for self-avoiding walks

The pivot algorithm is a Markov Chain Monte Carlo algorithm for simulating the self-avoiding walk. At each iteration a pivot which produces a global change in the walk is proposed. If the resulting walk is self-avoiding, the new walk is accepted; otherwise, it is rejected. Past implementations of the algorithm required a time O(N) per accepted pivot, where N is the number of steps in the walk. We show how to implement the algorithm so that the time required per accepted pivot is O(N^q) with q<1. We estimate that q is less than 0.57 in two dimensions, and less than 0.85 in three dimensions. Corrections to the O(N^q) make an accurate estimate of q impossible. They also imply that the asymptotic behavior of O(N^q) cannot be seen for walk lengths which can be simulated. In simulations the effective q is around 0.7 in two dimensions and 0.9 in three dimensions. Comparisons with simulations that use the standard implementation of the pivot algorithm using a hash table indicate that our implementation is faster by as much as a factor of 80 in two dimensions and as much as a factor of 7 in three dimensions. Our method does not require the use of a hash table and should also be applicable to the pivot algorithm for off-lattice models.

cond-mat↗

arXiv2000

Periodic Ground States in the Neutral Falicov-Kimball Model in Two Dimensions

We consider the Falicov-Kimball model in two dimensions in the neutral case, i.e, the number of mobile electrons is equal to the number of ions. For rational densities between 1/3 and 2/5 we prove that the ground state is periodic if the strength of the attraction between the ions and electrons is large enough. The periodic ground state is given by taking the one dimensional periodic ground state found by Lemberger and then extending it into two dimensions in such a way that the configuration is constant along lines at a 45 degree angle to the lattice directions.

cond-mat↗