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In many composites the electrical transport takes place only by tunneling between isolated particles. For a long time it was quite a puzzle how, in spite of the incompatibility of tunneling and percolation networks, these composites conform…

We study moderate deviations in the exponential corner growth model, both in the bulk setting and the increment-stationary setting. The main results are sharp right-tail bounds on the last-passage time and the exit point of the…

We consider an inhomogeneous oriented percolation model introduced by de Lima, Rolla and Valesin. In this model, the underlying graph is an oriented rooted tree in which each vertex points to each of its $d$ children with `short' edges, and…

We consider a one-dimensional random walk among biased i.i.d. conductances, in the case where the random walk is transient but sub-ballistic: this occurs when the conductances have a heavy-tail at $+\infty$ or at $0$. We prove that the…

We theoretically demonstrate a counter-intuitive phenomenon in optical interferometry with a thermal source: the emergence of second-order interference between two pairs of correlated optical paths even if the time delay imprinted by each…

A long standing open problem in extremal graph theory is to describe all graphs that maximize the number of induced copies of a path on four vertices. The character of the problem changes in the setting of oriented graphs, and becomes more…

In a randomly oriented graph containing vertices $x$ and $y$, denote by $\{x\to y\}$ the event that there is a directed path from $x$ to $y$. We study the correlation between the events $\{x\to y\}$ and $\{y\to z\}$ for a (large) oriented…

We show that the laws of scaling limits of nearcritical percolation exploration paths with different parameters are singular with respect to each other. This generalises a result of Nolin and Werner, using a similar technique. As a…

The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are shown to be invariant not only…

Recently Mertens and Moore [arXiv:1909.01484v1] showed that site percolation "is odd." By this they mean that on an $M\times N$ square lattice the number of distinct site configurations that allow for vertical percolation is odd. We report…

We derive a sufficient condition for the existence of a subcritical percolation phase for a wide range of continuum percolation models where each vertex is embedded into Euclidean space according to an iid-marked stationary Poisson point…

We introduce new probabilistic arguments to derive optimal-order central moment bounds in planar directed last-passage percolation. Our technique is based on couplings with the increment-stationary variants of the model, and is presented in…

Directed last passage percolation models on the plane, where one studies the weight as well as the geometry of optimizing paths (called polymers) in a field of i.i.d. weights, are paradigm examples of models in the KPZ universality class.…

We develop a new probabilistic method for deriving deviation estimates in directed planar polymer and percolation models. The key estimates are for exit points of geodesics as they cross transversal down-right boundaries. These bounds are…

We predict that self-bound clusters of particles exist in the supercritical phase of simple fluids. These clusters, whose internal temperature is lower than the global temperature of the system, define a percolation line that starts at the…

It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups $U(l)$, $Sp(2l)$ and $O(l)$. We present a theory of such…

Existing theory for multivariate extreme values focuses upon characterizations of the distributional tails when all components of a random vector, standardized to identical margins, grow at the same rate. In this paper, we consider the…

Correlation mixtures of elliptical copulas arise when the correlation parameter is driven itself by a latent random process. For such copulas, both penultimate and asymptotic tail dependence are much larger than for ordinary elliptical…

Osculating paths are sets of directed lattice paths which are not allowed to cross each other or have common edges, but are allowed to have common vertices. In this work we derive a constant term formula for the number of such lattice paths…

We define the model of two-dimensional random interlacements using simple random walk trajectories conditioned on never hitting the origin, and then obtain some properties of this model. Also, for random walk on a large torus conditioned on…