We study computational aspects of repulsive Gibbs point processes, which are probabilistic models of interacting particles in a finite-volume region of space. We introduce an approach for reducing a Gibbs point process to the hard-core model, a well-studied discrete spin system. Given an instance of such a point process, our reduction generates a random graph drawn from a natural geometric model. We show that the partition function of a hard-core model on graphs generated by the geometric model concentrates around the partition function of the Gibbs point process. Our reduction allows us to use a broad range of algorithms developed for the hard-core model to sample from the Gibbs point process and approximate its partition function. This is, to the extent of our knowledge, the first approach that deals with pair potentials of unbounded range.

We investigate the hyperuniformity of marked Gibbs point processes that have weak dependencies among distant points whilst the interactions of close points are kept arbitrary. Various stability and range assumptions are imposed on the Papangelou intensity in order to prove that the resulting point process is not hyperuniform. The scope of our results covers many frequently used models, including Gibbs point processes with a superstable, lower-regular, integrable pair potential, as well as the Widom–Rowlinson model with random radii and Gibbs point processes with interactions based on Voronoi tessellations and nearest-neighbour graphs.

We provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is superlinear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle (1970) for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite-range multi-body counterparts.

We show that a point process of hard spheres exhibits long-range orientational order. This process is designed to be a random perturbation of a three-dimensional lattice that satisfies a specific rigidity property; examples include the FCC and HCP lattices. We also define two-dimensional near-lattice processes by local geometry-dependent hard disk conditions. Earlier results about the existence of long-range orientational order carry over, and we obtain the existence of infinite-volume measures on two-dimensional point configurations that turn out to follow the orientation of a fixed triangular lattice arbitrarily closely.

The continuum random cluster model is a Gibbs modification of the standard Boolean model with intensity z > 0 and law of radii Q. The formal unnormalised density is given by q N cc , where q is a fixed parameter and N cc is the number of connected components in the random structure. We prove for a large class of parameters that percolation occurs for large enough z and does not occur for small enough z. We provide an application to the phase transition of the Widom–Rowlinson model with random radii. Our main tools are stochastic domination properties, a detailed study of the interaction of the model, and a Fortuin–Kasteleyn representation.

We study systems of $n$ points in the Euclidean space of dimension $d\geqslant 1$ interacting via a Riesz kernel $|x|^{-s}$ and confined by an external potential, in the regime where $d-2\leqslant s<d$. We also treat the case of logarithmic interactions in dimensions 1 and 2. Our study includes and retrieves all cases previously studied in Sandier and Serfaty [2D Coulomb gases and the renormalized energy, Ann. Probab. (to appear); 1D log gases and the renormalized energy: crystallization at vanishing temperature (2013)] and Rougerie and Serfaty [Higher dimensional Coulomb gases and renormalized energy functionals, Comm. Pure Appl. Math. (to appear)]. Our approach is based on the Caffarelli–Silvestre extension formula, which allows one to view the Riesz kernel as the kernel of an (inhomogeneous) local operator in the extended space $\mathbb{R}^{d+1}$.

As $n\rightarrow \infty$, we exhibit a next to leading order term in $n^{1+s/d}$ in the asymptotic expansion of the total energy of the system, where the constant term in factor of $n^{1+s/d}$ depends on the microscopic arrangement of the points and is expressed in terms of a ‘renormalized energy’. This new object is expected to penalize the disorder of an infinite set of points in whole space, and to be minimized by Bravais lattice (or crystalline) configurations. We give applications to the statistical mechanics in the case where temperature is added to the system, and identify an expected ‘crystallization regime’. We also obtain a result of separation of the points for minimizers of the energy.

Efficient algorithms for the simulation of strained heteroepitaxial growth with intermixing in 2+1 dimensions are presented. The first of these algorithms is an extension of the energy localization method [T. P. Schulze and P. Smereka, An energy localization principle and its application to fast kinetic Monte Carlo simulation of heteroepitaxial growth, J. Mech. Phys. Sol., 3 (2009), 521-538] from 1+1 to 2+1 dimensions. Two approximations of this basic algorithm are then introduced, one of which treats adatoms in a more efficient manner, while the other makes use of an approximation of the change in elastic energy in terms of local elastic energy density. In both cases, it is demonstrated that a reasonable level of fidelity is achieved. Results are presented showing how the film morphology is affected by misfit and deposition rate. In addition, simulations of stacked quantum dots are also presented.

We prove the existence of infinite-volume quermass-interaction processes in a general setting of nonlocally stable interaction and nonbounded convex grains. No condition on the parameters of the linear combination of the Minkowski functionals is assumed. The only condition is that the square of the random radius of the grain admits exponential moments for all orders. Our methods are based on entropy and large deviation tools.

We construct random dynamics for collections of nonintersecting planar contours, leaving invariant the distributions of length- and area-interacting polygonal Markov fields with V-shaped nodes. The first of these dynamics is based on the dynamic construction of consistent polygonal fields, as presented in the original articles by Arak (1983) and Arak and Surgailis (1989), (1991), and it provides an easy-to-implement Metropolis-type simulation algorithm. The second dynamics leads to a graphical construction in the spirit of Fernández et al. (1998), (2002) and yields a perfect simulation scheme in a finite window in the infinite-volume limit. This algorithm seems difficult to implement, yet its value lies in that it allows for theoretical analysis of the thermodynamic limit behaviour of length-interacting polygonal fields. The results thus obtained include, in the class of infinite-volume Gibbs measures without infinite contours, the uniqueness and exponential α-mixing of the thermodynamic limit of such fields in the low-temperature region. Outside this class, we conjecture the existence of an infinite number of extreme phases breaking both the translational and rotational symmetries.

In first-passage percolation (FPP) models, the passage time Tℓ from the origin to the point ℓeℓ satisfies f(ℓ) := ETℓ = μℓ + o(ℓ½+ε), where μ ∊ (0,∞) denotes the time constant. Yet, for lattice FPP, it is not known rigorously that f(ℓ) is eventually monotonically increasing. Here, for the Poisson-based Euclidean FPP of Howard and Newman (Prob. Theory Relat. Fields108 (1997), 153–170), we prove an explicit formula for f′(ℓ). In all dimensions, for certain values of the model's only parameter we have f′(ℓ) ≥ C > 0 for large ℓ.

In first-passage percolation models, the passage time T(0,L) from the origin to a point L is expected to exhibit deviations of order |L|χ from its mean, while minimizing paths are expected to exhibit fluctuations of order |L|ξ away from the straight line segment . Here, for Euclidean models in dimension d, we establish the lower bounds ξ ≥ 1/(d+1) and χ ≥(1-(d-1)ξ)/2. Combining this latter bound with the known upper bound ξ ≤ 3/4 yields that χ ≥ 1/8 for d=2.

We give an alternative proof of a point-process version of the FKG–Holley–Preston inequality which provides a sufficient condition for stochastic domination of probability measures, and for positive correlations of increasing functions.

Motivated by a statistical application, we consider continuum percolation in two or more dimensions, restricted to a large finite box, when above the critical point. We derive surface order large deviation estimates for the volume of the largest cluster and for its intersection with the boundary of the box. We also give some natural extensions to known, analogous results on lattice percolation.