Related papers: Permutons, meanders, and SLE-decorated Liouville q…

Liouville conformal field theory describes a random geometry that fluctuates around a deterministic one: the unique solution of the problem of finding, within a given conformal class, a Riemannian metric with prescribed scalar and geodesic…

Particular boundary correlation functions of conformal field theory are needed to answer some questions related to random conformally invariant curves known as Schramm-Loewner evolutions (SLE). In this article, we introduce a correspondence…

Let $(X_n)_{n \ge 0}$ be an irreducible, aperiodic, homogeneous Markov chain, with state space a totally ordered finite alphabet of size $m$. Using combinatorial constructions and weak invariance principles, we obtain the limiting shape of…

Two-dimensional loop-erased random walks (LERWs) are random planar curves whose scaling limit is known to be a Schramm-Loewner evolution SLE_k with parameter k = 2. In this note, some properties of an SLE_k trace on doubly-connected domains…

We consider uniform random permutations in proper substitution-closed classes and study their limiting behavior in the sense of permutons. The limit depends on the generating series of the simple permutations in the class. Under a mild…

We prove sandwich theorems and a Tauberian theorem in the space of compact metric measure spaces, endowed with the Gromov-Hausdorff-Prokhorov (GHP) topology. These results hold with respect to a close relative of Gromov's Lipschitz order.…

We propose variants of Schramm-Loewner evolution (SLE) that are related to superconformal algebras following the group theoretical formulation of SLE, in which the relevant stochastic differential equation is derived from a random process…

We prove that the Tutte embeddings (a.k.a. harmonic/barycentric embeddings) of certain random planar maps converge to $\gamma$-Liouville quantum gravity ($\gamma$-LQG). Specifically, we treat mated-CRT maps, which are discretized matings of…

What is the scaling limit of diffusion limited aggregation (DLA) in the plane? This is an old and famously difficult question. One can generalize the question in two ways: first, one may consider the {\em dielectric breakdown model}…

We describe the limit (for two topologies) of large uniform random square permutations, i.e., permutations where every point is a record. The starting point for all our results is a sampling procedure for asymptotically uniform square…

We establish a scaling limit result for the length $\operatorname{LIS}(\sigma_n)$ of the longest increasing subsequence of a permutation $\sigma_n$ of size $n$ sampled from the Brownian separable permuton $\boldsymbol{\mu}_p$ of parameter…

A large class of Jordan curves on the Riemann sphere can be encoded by circle homeomorphisms via conformal welding, among which we consider the welding homeomorphism of the random SLE loops and the Weil-Petersson class of quasicircles. It…

In their seminal paper introducing the theory of random graphs, Erd\H{o}s and R\'{e}nyi considered the evolution of the structure of a random subgraph of $K_n$ as the density increases from $0$ to $1$, identifying two key points in this…

We consider $m$ spinless Fermions in $l > m$ degenerate single-particle levels interacting via a $k$-body random interaction with Gaussian probability distribution and $k <= m$ in the limit $l$ to infinity (the embedded $k$-body random…

We construct the natural diffusion in the random geometry of planar Liouville quantum gravity. Formally, this is the Brownian motion in a domain $D$ of the complex plane for which the Riemannian metric tensor at a point $z \in D$ is given…

We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough…

We give an explicit construction of the scaling limit of the minimum spanning tree of the complete graph. The limit object is described using a recursive construction involving the convex minorants of a Brownian motion with parabolic drift…

For random collections of self-avoiding loops in two-dimensional domains, we define a simple and natural conformal restriction property that is conjecturally satisfied by the scaling limits of interfaces in models from statistical physics.…

The classical theorem by Pitman states that a Brownian motion minus twice its running infimum enjoys the Markov property. On the one hand, Biane understood that Pitman's theorem is intimately related to the representation theory of the…

We consider random genus-0 hyperbolic surfaces $\mathcal{S}_n$ with $n + 1$ punctures, sampled according to the Weil-Petersson measure. We show that, after rescaling the metric by $n^{-1/4}$, the surface $\mathcal{S}_n$ converges in…