For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop, and that outputs samples in exact accordance with the desired distribution. The method uses couplings, which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the al...

A major criticism of the statistical models for analyzing social networks developed by Holland, Leinhardt, and others wHolland, P.W., Leinhardt, S., 1977. Notes on the statistical analysis of social network data; Holland, P.W., Leinhardt, S., 1981. An exponential family of probability distributions for directed graphs. Journal of the American Statistical Association. 76, pp. 33–65 Ž with discussion.; Fienberg, S.E., Wasserman,

. We present a technique for simultaneously deriving two related results: Hydrodynamic scaling limits for one-dimensional asymmetric particle systems and asymptotic shapes for growth models. The idea is to specify the particle dynamics in terms of a microscopic Lax-Oleinik formula which leads directly to the macroscopic description in terms of a nonlinear conservation law. The law of large numbers required for this link comes from the growth model that is embedded in the particle system. In the limit, the asymptotic shape of the growth model becomes the convex conjugate of the flux of the conservation law, and the latter is computable from the particle system in equilibrium. The asymptotic shape is then obtained from the duality relation. The method is illustrated with four applications. Mathematics Subject Classification: Primary 60K35, Secondary 60C05, 82C22 Keywords: Hydrodynamic scaling limit, growth model, Lax-Oleinik formula, convex duality Address: Department of Mathematics, Iow...

We investigate the dynamics of ensembles of diffusive defects in one-dimensional deterministic cellular automata. The work builds on earlier results on individual random walks in c.a. (5;6) . Here we give a natural condition guaranteeing diffusive behavior also in the presence of other defects. Simple branching and birth mechanisms are introduced and prototype classes of cellular automata exhibiting weakly interacting walks capable of annihilation and coalescence are studied. Their equilibrium behavior is also characterized. The design principles of cellular automata with desired diffusive interaction properties becomes transparent from this analysis. Keywords: Cellular automaton, permutivity, topological defect, random walk. AMS Classification: 58F08, 60K35, 82C41 1 Research partially supported by the Academy of Finland and The Finnish Cultural Foundation 1 Introduction Topological defects, Bloch walls or contours can be identified in a number of standard lattice model...

this paper, we will study the limiting spatial structure of the voter model in d

We prove a hydrodynamic limit for ballistic deposition on a multidimensional lattice. In this growth model particles rain down at random and stick to the growing cluster at the first point of contact. The theorem is that if the initial random interface converges to a deterministic macroscopic function, then at later times the height of the scaled interface converges to the viscosity solution of a Hamilton-Jacobi equation. The proof idea is to decompose the interface into the shapes that grow from individual seeds of the initial interface. This decomposition converges to a variational formula that defines viscosity solutions of the macrosopic equation. The technical side of the proof involves subadditive methods and large deviation bounds for related first-passage percolation processes.

. We introduce a new stochastic representation of the partition function of the spin 1/2 Heisenberg ferromagnet. We express some of the relevant thermodynamic quantities in terms of expectations of functionals of so called random stirrings on Z d . By use of this representation we improve the lower bound on the pressure given by Conlon and Solovej in [CS2]. AMS subject classification (1991): 82D40, 82B20 1. Introduction and Result We consider the 1 2 -spin isotropic quantum Heisenberg ferromagnet (QHF) on the d-dimensional hypercubic lattice. The Hamiltonian of the model is H = 1 2 X jx\Gammayj=1 h (S(x) \Gamma S(y)) 2 \Gamma 1 i (1.1) where S(x) = (S X (x); S Y (x); S Z (x)) ; x 2 , are the local spin operators and the summation runs over nearest neighbour pairs of lattice sites in the rectangular box , with periodic boundary conditions. The canonical commutation relations satisfied by the spin operators are: \Theta S ff (x); S fi (y) = iffi x;y ffl ff;fi;fl S fl ...

We find limit shapes for a family of multiplicative measures on the set of partitions, induced by exponential generating functions with expansive parameters, ak ∼ Ck p−1, k → ∞, p> 0, where C is a positive constant. The measures considered are associated with the generalized Maxwell-Boltzmann models in statistical mechanics, reversible coagulation-fragmentation processes and combinatorial structures, known as assemblies. We prove a central limit theorem for fluctuations of a properly scaled partition chosen randomly according to the above measure, from its limit shape. We demonstrate that when the component size passes beyond the threshold value, the independence of numbers of components transforms into their conditional independence (given their masses). Among other things, the paper also discusses, in a general setting, the interplay between limit shape, threshold and gelation. 1

Random sampling has found numerous applications in computer science, statistics, and physics. The most widely applicable method of random sampling is to use a Markov chain whose steady state distribution is the probability distribution ß from which we wish to sample. After the Markov chain has been run for long enough, its state is approximately distributed according to ß. The principal problem with this approach is that it is often difficult to determine how long to run the Markov chain. In this thesis we present several algorithms that use Markov chains to return samples distributed exactly according to ß. The algorithms determine on their own how long to run the Markov chain. Two of the algorithms may be used with any Markov chain, but are useful only if the state space is not too large. Nonetheless, a spin-off of these two algorithms is a procedure for sampling random spanning trees of a directed graph that runs more quickly than the Aldous/Broder algorithm. Another of the exact sa...

Lanchier and Neuhauser have initiated the study of host-symbiont systems but have concentrated on the case in which the birth rates for unassociated hosts are equal. Here we allow the birth rates to be different and identify cases in which a host with a specialist pathogen can coexist with a second species. Our calculations suggest that it is possible for two hosts with specialist pathogens to coexist but it is not possible for a host with a specialist mutualist to coexist with a second species. 1