Related papers: Planar stochastic hyperbolic infinite triangulatio…

Many growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties. Such structures can often be described effectively by space-time trajectories of interacting…

We study the asymptotic behaviour of the most likely trajectories of a planar random walk that result in large deviations of the area of their convex hull. If the Laplace transform of the increments is finite on $R^2$, such a scaled limit…

We introduce an ensemble of infinite causal triangulations, called the uniform infinite causal triangulation, and show that it is equivalent to an ensemble of infinite trees, the uniform infinite planar tree. It is proved that in both cases…

One of the consequences of residual finiteness of triangle groups is that for any given hyperbolic triple $(\ell,m,n)$ there exist infinitely many regular hypermaps of type $(\ell,m,n)$ on compact orientable surfaces. The same conclusion…

As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a stochastic maximal inequality derived by using the Taylor expansion, is…

We study longest and shortest directed paths in the following natural model of directed random planar maps: the uniform infinite bipolar-oriented triangulation (UIBOT), which is the local limit of uniform bipolar-oriented triangulations…

Spatial random permutations were originally studied due to their connections to Bose-Einstein condensation, but they possess many interesting properties of their own. For random permutations of a regular lattice with periodic boundary…

Let N be a topologically finite, orientable 3-manifold with ideal triangulation. We show that if there is a solution to the hyperbolic gluing equations, then all edges in the triangulation are essential. This result is extended to a…

We show that a uniformly acute triangulation of the plane is rigid under Luo's discrete conformal change, extending previous results on hexagonal triangulations. Our result is a discrete analogue of the conformal rigidity of the plane. We…

In this paper we analyze local structure of several chaotic attractors recently suggested in literature as pseudohyperbolic. The absence of tangencies and thus the presence of the pseudohyperbolicity is verified using the method of angles…

We study the asymptotic behavior the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by [26]. We do this by obtaining bounds on the effective resistance between the…

We prove that uniform random triangulations whose genus is proportional to their size $n$ have diameter of order $\log n$ with high probability. We also show that in such triangulations, the distances between most pairs of points differ by…

We set the foundation for a series of works aimed at proving strong relations between uniform random planar maps and Liouville quantum gravity (LQG). Our method relies on a bijective encoding of site-percolated planar triangulations by…

The aim of this paper is to study asymptotic geometric properties almost surely or/and in probability of extreme order statistics of an i.i.d. random field (potential) indexed by sites of multidimensional lattice cube, the volume of which…

This paper concerns random bipartite planar maps which are defined by assigning weights to their faces. The paper presents a threefold contribution to the theory. Firstly, we prove the existence of the local limit for all choices of weights…

We consider a finite family of invertible $2 \times 2$ real matrices and a transitive Markov shift on the index set. Let $\lambda$ be the top Lyapunov exponent for random matrix products driven by the Markov shift. We prove that, if the…

We prove that in any recurrent reversible random rooted graph, two independent simple random walks started at the same vertex collide infinitely often almost surely. This applies to the Uniform Infinite Planar Triangulation and…

There are many classical random walk in random environment results that apply to ergodic random planar environments. We extend some of these results to random environments in which the length scale varies from place to place, so that the…

We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those…

The goal of this article is two-fold: in a first part, we prove Azuma-Hoeffding type concentration inequalities around the drift for the displacement of non-elementary random walks on hyperbolic spaces. For a proper hyperbolic space $M$, we…