Sublinearity of the travel-time variance for dependent first-passage percolation

Let $E$ be the set of edges of the $d$-dimensional cubic lattice $\mathbb{Z}^d$, with $d\geq2$, and let $t(e),e\in E$, be nonnegative values. The passage time from a vertex $v$ to a vertex $w$ is defined as $\inf_{\pi:v\rightarrow w}\sum_{e\in\pi}t(e)$, where the infimum is over all paths $\pi$ from $v$ to $w$, and the sum is over all edges $e$ of $\pi$. Benjamini, Kalai and Schramm [2] proved that if the $t(e)$'s are i.i.d. two-valued positive random variables, the variance of the passage time from the vertex 0 to a vertex $v$ is sublinear in the distance from 0 to $v$. This result was extended to a large class of independent, continuously distributed $t$-variables by Bena\"{\i}m and Rossignol [1]. We extend the result by Benjamini, Kalai and Schramm in a very different direction, namely to a large class of models where the $t(e)$'s are dependent. This class includes, among other interesting cases, a model studied by Higuchi and Zhang [9], where the passage time corresponds with the minimal number of sign changes in a subcritical "Ising landscape."Comment: Published in at http://dx.doi.org/10.1214/10-AOP631 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Developments in perfect simulation of Gibbs measures through a new result for the extinction of Galton-Watson-like processes

This paper deals with the problem of perfect sampling from a Gibbs measure with infinite range interactions. We present some sufficient conditions for the extinction of processes which are like supermartingales when large values are taken. This result has deep consequences on perfect simulation, showing that local modifications on the interactions of a model do not affect simulability. We also pose the question to optimize over a class of sequences of sets that influence the sufficient condition for the perfect simulation of the Gibbs measure. We completely solve this question both for the long range Ising models and for the spin models with finite range interactions.Comment: 28 page

Coupling and Bernoullicity in random-cluster and Potts models

An explicit coupling construction of random-cluster measures is presented. As one of the applications of the construction, the Potts model on amenable Cayley graphs is shown to exhibit at every temperature the mixing property known as Bernoullicity

Coupling from the past times with ambiguities and perturbations of interacting particle systems

We discuss coupling from the past techniques (CFTP) for perturbations of interacting particle systems on the d-dimensional integer lattice, with a finite set of states, within the framework of the graphical construction of the dynamics based on Poisson processes. We first develop general results for what we call CFTP times with ambiguities. These are analogous to classical coupling (from the past) times, except that the coupling property holds only provided that some ambiguities concerning the stochastic evolution of the system are resolved. If these ambiguities are rare enough on average, CFTP times with ambiguities can be used to build actual CFTP times, whose properties can be controlled in terms of those of the original CFTP time with ambiguities. We then prove a general perturbation result, which can be stated informally as follows. Start with an interacting particle system possessing a CFTP time whose definition involves the exploration of an exponentially integrable number of points in the graphical construction, and which satisfies the positive rates property. Then consider a perturbation obtained by adding new transitions to the original dynamics. Our result states that, provided that the perturbation is small enough (in the sense of small enough rates), the perturbed interacting particle system too possesses a CFTP time (with nice properties such as an exponentially decaying tail). The proof consists in defining a CFTP time with ambiguities for the perturbed dynamics, from the CFTP time for the unperturbed dynamics. Finally, we discuss examples of particle systems to which this result can be applied. Concrete examples include a class of neighbor-dependent nucleotide substitution model, and variations of the classical voter model, illustrating the ability of our approach to go beyond the case of weakly interacting particle systems.Comment: This paper is an extended and revised version of an earlier manuscript available as arXiv:0712.0072, where the results were limited to perturbations of RN+YpR nucleotide substitution model

How to Couple from the Past Using a Read-Once Source of Randomness

We give a new method for generating perfectly random samples from the stationary distribution of a Markov chain. The method is related to coupling from the past (CFTP), but only runs the Markov chain forwards in time, and never restarts it at previous times in the past. The method is also related to an idea known as PASTA (Poisson arrivals see time averages) in the operations research literature. Because the new algorithm can be run using a read-once stream of randomness, we call it read-once CFTP. The memory and time requirements of read-once CFTP are on par with the requirements of the usual form of CFTP, and for a variety of applications the requirements may be noticeably less. Some perfect sampling algorithms for point processes are based on an extension of CFTP known as coupling into and from the past; for completeness, we give a read-once version of coupling into and from the past, but it remains unpractical. For these point process applications, we give an alternative coupling method with which read-once CFTP may be efficiently used.Comment: 28 pages, 2 figure

Propp-Wilson algorithms and finitary codings for high noise Markov random fields

In this paper, we combine two previous works, the first being by the first author and K. Nelander, and the second by J. van den Berg and the second author, to show (1) that one can carry out a Propp--Wilson exact simulation for all Markov random fields on Z d satisfying a certain high noise assumption, and (2) that all such random fields are a finitary image of a finite state i.i.d. process. (2) is a strengthening of the previously known fact that such random fields are so-called Bernoulli shifts. 1 Introduction A random field with finite state space S indexed by the integer lattice Z d is a random mapping X : Z d ! S, or it can equivalently be seen as a random element of S Z d . Here we focus on so-called Markov random fields, characterized by having a dependency structure which only propagates via interactions between nearest neighbors in Z d . We specialize further to Markov random fields satsifying a certain high noise assumption, which says that these interactions shou..