We consider an Ising spin system with Kac potentials in a torus of $Z^d$, $d\ge$2, and fix the temperature below its Lebowitz-Penrose critical value. We prove that when the Kac scaling parameter $\gamma$ vanishes, the log of the probability of an interface becomes proportional to its area and the surface tension, related to the proportionality constant, converges to the van der Waals surface tension. The results are based on the analysis of the rate functionals for Gibbsian large deviations and on the proof that they Gamma-converge to the perimeter functional of geometric measure theory (which extends the notion of area). Our considerations include non smooth interfaces proving that the Gibbsian probability of an interface depends only on its area and not on its regularity.
Surface tension in Ising systems with Kac potential
ALBERTI, GIOVANNI;
1996-01-01
Abstract
We consider an Ising spin system with Kac potentials in a torus of $Z^d$, $d\ge$2, and fix the temperature below its Lebowitz-Penrose critical value. We prove that when the Kac scaling parameter $\gamma$ vanishes, the log of the probability of an interface becomes proportional to its area and the surface tension, related to the proportionality constant, converges to the van der Waals surface tension. The results are based on the analysis of the rate functionals for Gibbsian large deviations and on the proof that they Gamma-converge to the perimeter functional of geometric measure theory (which extends the notion of area). Our considerations include non smooth interfaces proving that the Gibbsian probability of an interface depends only on its area and not on its regularity.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
