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DDC: 510 × Subject: cluster expansion ×

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Now showing 1 - 3 of 3
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    Gibbs point processes on path space: Existence, cluster expansion and uniqueness
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Zass, Alexander
    We study a class of infinite-dimensional diffusions under Gibbsian interactions, in the context of marked point configurations: The starting points belong to R^d, and the marks are the paths of Langevin diffusions. We use the entropy method to prove existence of an infinite-volume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds.
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    Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Bovier, Anton; Hollander, Frank den; Spitoni, Cristian
    In this paper we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let $b$ denote the inverse temperature and let $L_b subset Z^2$ be a square box with periodic boundary conditions such that $lim_btoinfty L_b =infty$. We run the dynamics on $L_b$ starting from a random initial configuration where all the droplets (= clusters of plus-spins, respectively, clusters of particles) are small. For large $b$, and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single emphcritical droplet somewhere in $L_b$. Using potential-theoretic methods, we compute the emphaverage nucleation time (= the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to one as $btoinfty$. It turns out that this time grows as $Ke^Gammab/ L_b $ for Glauber dynamics and $Kb e^Gammab/ L_b $ for Kawasaki dynamics, where $Gamma$ is the local canonical, respectively, grand-canonical energy to create a critical droplet and $K$ is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on $ L_b $). The fact that the average nucleation time is inversely proportional to $ L_b $ is referred to as emphhomogeneous nucleation, because it says that the critical droplet for the transition appears essentially independently in small boxes that partition $L_b$.
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    Gibbsianness of locally thinned random fields
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Engler, Nils; Jahnel, Benedikt; Külske, Christof
    We consider the locally thinned Bernoulli field on ℤ d, which is the lattice version of the Type-I Matérn hardcore process in Euclidean space. It is given as the lattice field of occupation variables, obtained as image of an i.i.d. Bernoulli lattice field with occupation probability p, under the map which removes all particles with neighbors, while keeping the isolated particles. We prove that the thinned measure has a Gibbsian representation and provide control on its quasilocal dependence, both in the regime of small p, but also in the regime of large p, where the thinning transformation changes the Bernoulli measure drastically. Our methods rely on Dobrushin uniqueness criteria, disagreement percolation arguments [46], and cluster expansions