The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here main results that have been obtained, both in two and higher dimensions. In particular, we describe how phenomenological Wulff Winterbottom constructions can be derived from description provided by statistical mechanics lattice gases. focus on conceptual issues central ideas existing approaches.

In this paper we study the metastable behavior of one simplest disordered spin system, random field Curie-Weiss model. We will show how potential theoretic approach can be used to prove sharp estimates on capacities and exit times also in case when distribution is continuous. Previous work was restricted takes only finitely many values, which allowed reduction a finite dimensional problem using lumping techniques. Here produce first genuine context where entropy important.

We develop a fluctuation theory of connectivities for subcritical random cluster models. The is based on comprehensive nonperturbative probabilistic description long connected clusters in terms essentially one-dimensional chains irreducible objects. Statistics local observables, example, displacement, over such obey classical limit laws, and our construction leads to an effective walk representation percolation clusters. results include derivation sharp Ornstein–Zernike type asymptotic...

In this paper, we develop a detailed analysis of critical prewetting in the context two-dimensional Ising model. Namely, consider nearest-neighbor model 2N×N rectangular box with boundary condition inducing coexistence + phase bulk and layer − along bottom wall. The presence an external magnetic field intensity h=λ/N (for some fixed λ>0) makes unstable. For any β>βc, prove that, under diffusing scaling by N−2/3 horizontally N−1/3 vertically, interface separating unstable from weakly...

We derive a precise Ornstein–Zernike asymptotic formula for the decay of two-point function $\mathbb{P}_p (0 \leftrightarrow x)$ Bernoulli bond percolation on integer lattice $\mathbb{Z}^d$ in any dimension $d \geq 2$, direction $x$ and subcritical value $p < p_c (d)$.

We prove invariance principles for phase separation lines in the two dimensional nearest neighbour Ising model up to critical temperature and connectivity general context of high finite range ferromagnetic models.

We describe some recent results concerning the statistical properties of a self-interacting polymer stretched by an external force. concentrate mainly on cases purely attractive or repulsive self-interactions, but our are stable under suitable small perturbations these pure cases. provide in particular precise description phase (local limit theorems for endpoint and local observables, invariance principle, microscopic structure). Our also characterize precisely (nontrivial,...

We consider a model of polymer in $\mathbb{Z}^{d+1}$, constrained to join 0 and hyperplane at distance $N$. The is subject quenched nonnegative random environment. Alternatively, the describes crossing walks potential (see Zerner [Ann Appl. Probab. 8 (1998) 246--280] or Chapter 5 Sznitman [Brownian Motion, Obstacles Random Media Springer] for original Brownian motion formulation). It was recently shown [Ann. 36 (2008) 1528--1583; Theory Related Fields 143 (2009) 615--642] that, such setting,...

We consider confinement properties of families non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials tilted area type. The model is introduced in order mimic level lines $2+1$ discrete Solid-On-Solid random interfaces wall.