arXiv subjects

Leslie Ann Goldberg

Publications and source records attributed to Leslie Ann Goldberg.

94 records · Page 6Linked to original sources

arXiv2006

Markov chain comparison

This is an expository paper, focussing on the following scenario. We have two Markov chains, $\mathcal {M}$ and $\mathcal {M}'$. By some means, we have obtained a bound on the mixing time of $\mathcal {M}'$. We wish to compare $\mathcal {M}$ with $\mathcal {M}'$ in order to derive a corresponding bound on the mixing time of $\mathcal {M}$. We investigate the application of the comparison method of Diaconis and Saloff-Coste to this scenario, giving a number of theorems which characterize the applicability of the method. We focus particularly on the case in which the chains are not reversible. The purpose of the paper is to provide a catalogue of theorems which can be easily applied to bound mixing times.

math.PR↗

arXiv2006

Systematic scan for sampling colorings

We address the problem of sampling colorings of a graph $G$ by Markov chain simulation. For most of the article we restrict attention to proper $q$-colorings of a path on $n$ vertices (in statistical physics terms, the one-dimensional $q$-state Potts model at zero temperature), though in later sections we widen our scope to general ``$H$-colorings'' of arbitrary graphs $G$. Existing theoretical analyses of the mixing time of such simulations relate mainly to a dynamics in which a random vertex is selected for updating at each step. However, experimental work is often carried out using systematic strategies that cycle through coordinates in a deterministic manner, a dynamics sometimes known as systematic scan. The mixing time of systematic scan seems more difficult to analyze than that of random updates, and little is currently known. In this article we go some way toward correcting this imbalance. By adapting a variety of techniques, we derive upper and lower bounds (often tight) on the mixing time of systematic scan. An unusual feature of systematic scan as far as the analysis is concerned is that it fails to be time reversible.

math.PR↗

arXiv2005

Improved mixing bounds for the anti-ferromagnetic Potts model on Z^2

We consider the anti-ferromagnetic Potts model on the the integer lattice Z^2. The model has two parameters, q, the number of spins, and λ=\exp(-β), where βis ``inverse temperature''. It is known that the model has strong spatial mixing if q>7 or if q=7 and λ=0 or λ> 1/8 or if q=6 and λ=0 or λ> 1/4. The lambda=0 case corresponds to the model in which configurations are proper q-colourings of Z^2. We show that the system has strong spatial mixing for q >= 6 and any λ. This implies that Glauber dynamics is rapidly mixing (so there is a fully-polynomial randomised approximation scheme for the partition function) and also that there is a unique infinite-volume Gibbs state. We also show that strong spatial mixing occurs for a larger range of λthan was previously known for q = 3, 4 and 5.

math-ph↗

arXiv2005

Utilitarian resource assignment

This paper studies a resource allocation problem introduced by Koutsoupias and Papadimitriou. The scenario is modelled as a multiple-player game in which each player selects one of a finite number of known resources. The cost to the player is the total weight of all players who choose that resource, multiplied by the ``delay'' of that resource. Recent papers have studied the Nash equilibria and social optima of this game in terms of the $L_\infty$ cost metric, in which the social cost is taken to be the maximum cost to any player. We study the $L_1$ variant of this game, in which the social cost is taken to be the sum of the costs to the individual players, rather than the maximum of these costs. We give bounds on the size of the coordination ratio, which is the ratio between the social cost incurred by selfish behavior and the optimal social cost; we also study the algorithmic problem of finding optimal (lowest-cost) assignments and Nash Equilibria. Additionally, we obtain bounds on the ratio between alternative Nash equilibria for some special cases of the problem.

cs.GT↗