We consider Poisson hail models and characterize up to boundaries the collection of critical moments which guarantee stability. In particular, we treat the case of infinite speed of propagation.

We study Bose gases in $d \ge 2$ dimensions with short-range repulsive pair interactions at positive temperature, in the canonical ensemble and in the thermodynamic limit. We assume the presence of hard Poissonian obstacles and focus on the non-percolation regime. For sufficiently strong interparticle interactions, we show that almost surely there cannot be Bose–Einstein condensation into a sufficiently localized, normalized one-particle state. The results apply to the canonical eigenstates of the underlying one-particle Hamiltonian.

The Parisi formula is a self-contained description of the infinite-volume limit of the free energy of mean-field spin glass models. We showthat this quantity can be recast as the solution of a Hamilton–Jacobi equation in the Wasserstein space of probability measures on the positive half-line.

In this paper we propose a polynomial-time deterministic algorithm for approximately counting the k-colourings of the random graph G(n, d/n), for constant d>0. In particular, our algorithm computes in polynomial time a $(1\pm n^{-\Omega(1)})$ -approximation of the so-called ‘free energy’ of the k-colourings of G(n, d/n), for $k\geq (1+\varepsilon) d$ with probability $1-o(1)$ over the graph instances.

Our algorithm uses spatial correlation decay to compute numerically estimates of marginals of the Gibbs distribution. Spatial correlation decay has been used in different counting schemes for deterministic counting. So far algorithms have exploited a certain kind of set-to-point correlation decay, e.g. the so-called Gibbs uniqueness. Here we deviate from this setting and exploit a point-to-point correlation decay. The spatial mixing requirement is that for a pair of vertices the correlation between their corresponding configurations becomes weaker with their distance.

Furthermore, our approach generalizes in that it allows us to compute the Gibbs marginals for small sets of nearby vertices. Also, we establish a connection between the fluctuations of the number of colourings of G(n, d/n) and the fluctuations of the number of short cycles and edges in the graph.

The effect of disorder for pinning models is a subject which has attracted much attention in theoretical physics and rigorous mathematical physics. A peculiar point of interest is the question of coincidence of the quenched and annealed critical point for a small amount of disorder. The question has been mathematically settled in most cases in the last few years, giving in particular a rigorous validation of the Harris criterion on disorder relevance. However, the marginal case, where the return probability exponent is equal to $1/2$, that is, where the interarrival law of the renewal process is given by $\text{K}(n)=n^{-3/2}\unicode[STIX]{x1D719}(n)$ where $\unicode[STIX]{x1D719}$ is a slowly varying function, has been left partially open. In this paper, we give a complete answer to the question by proving a simple necessary and sufficient criterion on the return probability for disorder relevance, which confirms earlier predictions from the literature. Moreover, we also provide sharp asymptotics on the critical point shift: in the case of the pinning of a one-dimensional simple random walk, the shift of the critical point satisfies the following high temperature asymptotics

$$\begin{eqnarray}\lim _{\unicode[STIX]{x1D6FD}\rightarrow 0}\unicode[STIX]{x1D6FD}^{2}\log h_{c}(\unicode[STIX]{x1D6FD})=-\frac{\unicode[STIX]{x1D70B}}{2}.\end{eqnarray}$$

We consider an independent long-range bond percolation on Z2. Horizontal and vertical bonds of length n are independently open with probability p_n ∈ [0, 1]. Given ∑n=1∞∏i=1n(1 − pi) < ∞, we prove that there exists an infinite cluster of open bonds of length less than or equal to N for some large but finite N. The result gives a partial answer to the truncation problem.

We study ergodic random Schrödinger operators on a covering manifold, where the randomness enters both via the potential and the metric. We prove measurability of the random operators, almost sure constancy of their spectral properties, the existence of a self-averaging integrated density of states and a Pastur–šubin type trace formula.

We introduce and study a novel type of first-passage percolation problem on where the associated first-passage time measures the density of interface between two types of sites. If the types, designated + and –, are independently assigned their values with probability p and (1 — p) respectively, we show that the density of interface is non-zero provided that both species are subcritical with regard to percolation, i.e. pc > p > 1 – pc. Furthermore, we show that as p ↑ pc or p ↓ (1 – pc), the interface density vanishes with scaling behavior identical to the correlation length of the site percolation problem.