Results 1  10
of
18
Exact Sampling with Coupled Markov Chains and Applications to Statistical Mechanics
, 1996
"... 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 ..."
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Cited by 411 (13 self)
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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...
What do we know about the Metropolis algorithm
 J. Comput. System. Sci
, 1998
"... The Metropolis algorithm is a widely used procedure for sampling from a specified distribution on a large finite set. We survey what is rigorously known about running times. This includes work from statistical physics, computer science, probability and statistics. Some new results are given ae an il ..."
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Cited by 62 (7 self)
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The Metropolis algorithm is a widely used procedure for sampling from a specified distribution on a large finite set. We survey what is rigorously known about running times. This includes work from statistical physics, computer science, probability and statistics. Some new results are given ae an illustration of the geometric theory of Markov chains. 1. Introduction. Let % be a finite set and m(~)> 0 a probability distribution on %. The Metropolis algorithm is a procedure for drawing samples from X. It was introduced by Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller [1953]. The algorithm requires the user to specify a connected, aperiodic Markov chain 1<(z, y) on %. This chain need not be symmetric but if K(z, y)>0, one needs 1<(Y, z)>0. The chain K is modified
metastability threshold for twodimensional bootstrap percolation
 Probab. Theory Related Fields 125 (2003
"... In the bootstrap percolation model, sites in an L by L square are initially independently declared active with probability p. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as p → 0 and L → ∞ simultaneously of the probabil ..."
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Cited by 39 (7 self)
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In the bootstrap percolation model, sites in an L by L square are initially independently declared active with probability p. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as p → 0 and L → ∞ simultaneously of the probability I(L,p) that the entire square is eventually active. We prove that I(L,p) → 1 if lim inf p log L> λ, and I(L,p) → 0 if lim sup p log L < λ, where λ = π 2 /18. We prove the same behaviour, with the same threshold λ, for the probability J(L,p) that a site is active by time L in the process on the infinite lattice. The same results hold for the socalled modified bootstrap percolation model, but with threshold λ ′ = π 2 /6. The existence of the thresholds λ,λ ′ settles a conjecture of Aizenman and Lebowitz [3], while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony [2]. 1
Wulff droplets and the metastable relaxation of kinetic Ising models
 Comm. Math. Phys
, 1998
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For 2D lattice spin systems Weak Mixing Implies Strong Mixing
"... . We prove that for finite range discrete spin systems on the two dimensional lattice Z 2 , the (weak) mixing condition which follows, for instance, from the DobrushinShlosman uniqueness condition for the Gibbs state implies a stronger mixing property of the Gibbs state, similar to the DobrushinS ..."
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Cited by 31 (5 self)
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. We prove that for finite range discrete spin systems on the two dimensional lattice Z 2 , the (weak) mixing condition which follows, for instance, from the DobrushinShlosman uniqueness condition for the Gibbs state implies a stronger mixing property of the Gibbs state, similar to the DobrushinShlosman complete analiticity condition, but restricted to all squares in the lattice, or, more generally, to all sets multiple of a large enough square. The key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk, because of the weak mixing condition, or along the boundary because it is one dimensional. As a consequence we obtain for ferromagnetic Isingtype systems proofs that several nice properties hold arbitrarily close to the critical temperature; these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic Sobolev constant and rapid convergence to equilibrium of the assoc...
On the Two Dimensional Dynamical Ising Model In the Phase Coexistence Region
 J. Stat. Phys
, 1994
"... . We consider a Glauber dynamics reversible with respect to the two dimensional Ising model in a finite square of side L, in the absence of an external field and at large inverse temperature fi. We first consider the gap in the spectrum of the generator of the dynamics in two different cases: with p ..."
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Cited by 18 (9 self)
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. We consider a Glauber dynamics reversible with respect to the two dimensional Ising model in a finite square of side L, in the absence of an external field and at large inverse temperature fi. We first consider the gap in the spectrum of the generator of the dynamics in two different cases: with plus and open boundary condition. We prove that, when the symmetry under global spin flip is broken by the boundary conditions, the gap is much larger than the case in which the symmetry is present. For this latter we compute exactly the asymptotics of \Gamma 1 fiL log(gap) as L ! 1 and show that it coincides with the surface tension along one of the coordinat axes. As a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spinspin time correlation in the infinite volume plus phase. Our results establish a connection between the dynamical large deviations and those of the equilibrium Gibbs measure studied by...
Rare Events in Stochastic Partial Differential Equations on Large Spatial Domains
"... A methodology is proposed for studying rare events in stochastic partial differential equations in systems that are so large that standard large deviation theory does not apply. The idea is to deduce the behavior of the original model by breaking the system into appropriately scaled subsystems that ..."
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Cited by 7 (1 self)
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A methodology is proposed for studying rare events in stochastic partial differential equations in systems that are so large that standard large deviation theory does not apply. The idea is to deduce the behavior of the original model by breaking the system into appropriately scaled subsystems that are sufficiently small for large deviation theory to apply but sufficiently large to be asymptotically independent from one another. The methodology is illustrated in the context of a simple onedimensional stochastic partial differential equation. The application reveals a connection between the dynamics of the partial differential equation and the classical Johnson–Mehl–Avrami–Kolmogorov nucleation and growth model. It also illustrates that rare events are much more likely and predictable in large systems than in small ones due to the extra entropy provided by space.
The spectral gap of the 2D stochastic Ising model with mixed boundary conditions
 J. Statist. Phys
"... Abstract. We establish upper bounds for the spectral gap of the stochastic Ising model at low temperature in an N ×N box, with boundary conditions which are “plus ” except for small regions at the corners which are either free or “minus. ” The spectral gap decreases exponentially in the size of the ..."
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Cited by 6 (1 self)
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Abstract. We establish upper bounds for the spectral gap of the stochastic Ising model at low temperature in an N ×N box, with boundary conditions which are “plus ” except for small regions at the corners which are either free or “minus. ” The spectral gap decreases exponentially in the size of the corner regions, when these regions are of size at least of order log N. This means that removing as few as O(log N) plus spins from the corners produces a spectral gap far smaller than the order N −2 gap believed to hold under the allplus boundary condition. Our results are valid at all subcritical temperatures.
Metastability And Typical Exit Paths In Stochastic Dynamics
, 1997
"... In this paper we review and discuss results on metastability. We consider ergodic aperiodic Markov chains with exponentially small transition probabilities and we give a complete description of the typical tube of trajectories during the first excursion outside a general domain Q. 25=aprile=1997 [1] ..."
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Cited by 5 (3 self)
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In this paper we review and discuss results on metastability. We consider ergodic aperiodic Markov chains with exponentially small transition probabilities and we give a complete description of the typical tube of trajectories during the first excursion outside a general domain Q. 25=aprile=1997 [1] 0:1 Section 1. Introduction. In this review we want to describe some recent results in the theory of metastability from the point of view of mathematical physics and probability theory. Mathematically we will deal with the study of some large deviation problems for families of Markov chains with finite state space and transition probabilities decaying exponentially fast in a large external parameter fi: the so called FreidlinWentzell regime. From physical point of view we analyze a central problem arising in the dynamical description of phase transitions: the decay from metastable to stable equilibrium. Phase transitions (like, for instance, liquidvapour for a fluid or positivenegativ...