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18
LARGEDEVIATIONS/THERMODYNAMIC APPROACH TO PERCOLATION ON THE COMPLETE GRAPH
"... ABSTRACT. We present a largedeviations/thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the largedeviation rate function for the probability that the giant component occupies a fixed fraction of the graph. One consequence is an immediat ..."
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ABSTRACT. We present a largedeviations/thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the largedeviation rate function for the probability that the giant component occupies a fixed fraction of the graph. One consequence is an immediate derivation of the “cavity ” formula for the fraction of sites in the giant component. As a byproduct of our analysis we compute also the largedeviation rate functions for the probabilities of the event that the random graph is connected, the event that it contains no loops and the event that it contains only “small ” components. 1.
STABILITY OF INTERFACES AND STOCHASTIC DYNAMICS IN THE REGIME OF PARTIAL WETTING.
"... Abstract. The goal of this paper is twofold. First, assuming strict convexity of the surface tension, we derive a stability property with respect to the Hausdorff distance of a coarse grained representation of the interface between the two pure phases of the Ising model. This improves the L 1 descri ..."
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Cited by 5 (3 self)
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Abstract. The goal of this paper is twofold. First, assuming strict convexity of the surface tension, we derive a stability property with respect to the Hausdorff distance of a coarse grained representation of the interface between the two pure phases of the Ising model. This improves the L 1 description of phase segregation. ✠Using this result and an additional assumption on mixing properties of the underlying FK measures, we are then able to derive bounds on the decay of the spectral gap of the Glauber dynamics in dimensions larger or equal to three. These bounds are related to previous results by Martinelli [Ma] in the twodimensional case. Our assumptions can be easily verified for low enough temperatures and, presumably, hold true in the whole of the phase coexistence region. ✠ 1.
A proof of the GibbsThomson formula in the droplet formation regime
, 2004
"... We study equilibrium droplets in twophase systems at parameter values corresponding to phase coexistence. Specifically, we give a selfcontained microscopic ..."
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Cited by 4 (2 self)
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We study equilibrium droplets in twophase systems at parameter values corresponding to phase coexistence. Specifically, we give a selfcontained microscopic
Droplet minimizers for the Cahn–Hilliard free energy functional
 J. Geometric Analysis
, 2006
"... ABSTRACT. We prove theorems characterizing the minimizers for the CahnHilliard free energy functional, which is used to describe the liquid vapor phase transition (or the 2 state magnetization transition). In particular, we exactly determine the critical density for droplet formation, and the geome ..."
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ABSTRACT. We prove theorems characterizing the minimizers for the CahnHilliard free energy functional, which is used to describe the liquid vapor phase transition (or the 2 state magnetization transition). In particular, we exactly determine the critical density for droplet formation, and the geometry of the droplets. 1.1. The variational problem 1.
Comment on: “Theory of the evaporation/condensation transition of equilibrium droplets in finite volumes”, Physica A (to appear
"... transition of equilibrium droplets in finite volumes” ..."
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Cited by 3 (3 self)
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transition of equilibrium droplets in finite volumes”
Colligative properties of solutions: I. Fixed concentrations
, 2005
"... Using the formalism of rigorous statistical mechanics, we study the phenomena of phase separation and freezingpoint depression upon freezing of solutions. Specifically, we devise an Isingbased model of a solvent–solute system and show that, in the ensemble with a fixed amount of solute, a macrosco ..."
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Using the formalism of rigorous statistical mechanics, we study the phenomena of phase separation and freezingpoint depression upon freezing of solutions. Specifically, we devise an Isingbased model of a solvent–solute system and show that, in the ensemble with a fixed amount of solute, a macroscopic phase separation occurs in an interval of values of the chemical potential of the solvent. The boundaries of the phase separation domain in the phase diagram are characterized and shown to asymptotically agree with the formulas used in heuristic analyses of freezingpoint depression. The limit of infinitesimal concentrations is described in a subsequent paper.
Colligative Properties of Solutions: II. Vanishing Concentrations
, 2005
"... We continue our study of colligative properties of solutions initiated in ref. 1. We focus on the situations where, in a system of linear size L, the concentration and the chemical potential scale like c = ξ/L and h = b/L, respectively. We find that there exists a critical value ξt such that no phas ..."
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We continue our study of colligative properties of solutions initiated in ref. 1. We focus on the situations where, in a system of linear size L, the concentration and the chemical potential scale like c = ξ/L and h = b/L, respectively. We find that there exists a critical value ξt such that no phase separation occurs for ξ � ξt while, for ξ>ξt, the two phases of the solvent coexist for an interval of values of b. Moreover, phase separation begins abruptly in the sense that a macroscopic fraction of the system suddenly freezes (or melts) forming a crystal (or droplet) of the complementary phase when b reaches a critical value. For certain values of system parameters, under “frozen ” boundary conditions, phase separation also ends abruptly in the sense that the equilibrium droplet grows continuously with increasing b and then suddenly jumps in size to subsume the entire system. Our findings indicate that the onset of freezingpoint depression is in fact a surface phenomenon.
ACMAC’s PrePrint Repository aim is to enable open access to the scholarly output of ACMAC.
, 2011
"... Zerorange processes with decreasing jump rates exhibit a condensation transition, where a positive fraction of all particles condenses on a single lattice site when the total density exceeds a critical value. We study the onset of condensation, i.e. the behaviour of the maximum occupation number af ..."
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Zerorange processes with decreasing jump rates exhibit a condensation transition, where a positive fraction of all particles condenses on a single lattice site when the total density exceeds a critical value. We study the onset of condensation, i.e. the behaviour of the maximum occupation number after adding or subtracting a subextensive excess mass of particles at the critical density. We establish a law of large numbers for the excess mass fraction in the maximum, which turns out to jump from zero to a positive value at a critical scale. Our results also include distributional limits for the fluctuations of the maximum, which change from standard extreme value statistics to Gaussian when the density crosses the critical point. Fluctuations in the bulk are also covered, showing that the mass outside the maximum is distributed homogeneously. In summary, we identify the detailed behaviour at the critical scale including subleading terms, which provides a full understanding of the crossover from sub to supercritical behaviour.