相关论文: Separating cycles and isoperimetric inequalities i…

We introduce and study the uniform infinite planar quadrangulation (UIPQ) with a boundary via an extension of the construction of arXiv:1201.1052. We then relate this object to its simple boundary analog using a pruning procedure. This…

We consider the model of the Brownian plane, which is a pointed non-compact random metric space with the topology of the complex plane. The Brownian plane can be obtained as the scaling limit in distribution of the uniform infinite planar…

We give a new construction of the uniform infinite half-planar quadrangulation with a general boundary (or UIHPQ), analogous to the construction of the UIPQ presented by Chassaing and Durhuus, which allows us to perform a detailed study of…

We introduce a new construction of the Uniform Infinite Planar Quadrangulation (UIPQ). Our approach is based on an extension of the Cori-Vauquelin-Schaeffer mapping in the context of infinite trees, in the spirit of previous work. However,…

The uniform infinite planar quadrangulation is an infinite random graph embedded in the plane, which is the local limit of uniformly distributed finite quadrangulations with a fixed number of faces. We study asymptotic properties of this…

We consider the uniform infinite quadrangulation of the plane (UIPQ). Curien, M\'enard and Miermont recently established that in the UIPQ, all infinite geodesic rays originating from the root are essentially similar, in the sense that they…

A construction as a growth process for sampling of the uniform infinite planar triangulation (UIPT), defined in a previous paper, is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the…

We study large uniform random quadrangulations whose genus grow linearly with the number of faces, whose local convergence was recently established by Budzinski and the author arXiv:1902.00492,arXiv:2012.05813. Here we study several…

We establish a connection between the uniform infinite planar triangulation and some critical time-reversed branching process. This allows to find a scaling limit for the principal boundary component of a ball of radius R for large R (i.e.…

We study non-compact scaling limits of uniform random planar quadrangulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe…

We study an annealed model of Uniform Infinite Planar Quadrangulation (UIPQ) with an infinite two-sided self-avoiding walk (SAW), which can also be described as the result of glueing together two independent uniform infinite…

We consider the unique infinite connected component of supercritical bond percolation on the square lattice and study the geometric properties of isoperimetric sets, i.e., sets with minimal boundary for a given volume. For almost every…

We identify the local scaling limit of the Uniform Infinite Planar Quadrangulation (UIPQ) and of critical Boltzmann quadrangulations, when one simultaneously rescales the distances and reroot them far away from the root of a distinguished…

We construct the uniform infinite planar map (UIPM), obtained as the n \to \infty local limit of planar maps with n edges, chosen uniformly at random. We then describe how the UIPM can be sampled using a "peeling" process, in a similar way…

We study the scaling limit of the volume and perimeter of the discovered regions in the Markovian explorations known as peeling processes for infinite random planar maps such as the uniform infinite planar triangulation (UIPT) or…

We introduce a one-parameter family of random infinite quadrangulations of the half-plane, which we call the uniform infinite half-planar quadrangulations with skewness (UIHPQ$_p$ for short, with $p\in[0,1/2]$ measuring the skewness). They…

We prove that geodesic rays in the Uniform Infinite Planar Triangulation (UIPT) coalesce in a strong sense using the skeleton decomposition of random triangulations discovered by Krikun. This implies the existence of a unique horofunction…

We prove that any distributional limit of finite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent. As a corollary, we obtain that the uniform infinite planar triangulation and quadrangulation…

Pursuing the approach of Angel & Ray, we introduce and study a family of random infinite triangulations of the full-plane that satisfy a natural spatial Markov property. These new random lattices naturally generalize Angel & Schramm's…

The discrete isoperimetric inequality in Euclidean geometry states that among all $n$-gons having a fixed perimeter $p$, the one with the largest area is the regular $n$-gon. The statement is true in spherical geometry and hyperbolic…