Related papers: On a surface formed by randomly gluing together po…

We investigate the asymptotic properties of random polytopes arising as convex hulls of $n$ independent random points sampled from a family of block-beta distributions. Notably, this family includes the uniform distribution on a product of…

For a compact surface $S$, let $\mathcal{I}(S)$ denote the Torelli group of $S$. For a compact orientable surface $\Sigma$, $\mathcal{I}(\Sigma)$ is generated by BSCC maps and BP maps. For a non-orientable closed surface $N$,…

We introduce and study a universal model of random geometry in two dimensions. To this end, we start from a discrete graph drawn on the sphere, which is chosen uniformly at random in a certain class of graphs with a given size $n$, for…

Let $K_n$ be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on $\R^d$. We establish variance asymptotics as $n \to \infty$ for the re-scaled intrinsic volumes and $k$-face functionals of…

In this paper we deduce a universal result about the asymptotic distribution of roots of random polynomials, which can be seen as a complement to an old and famous result of Erdos and Turan. More precisely, given a sequence of random…

We classify homomorphisms from the braid group on $n$ strands to the pure mapping class group of a nonoriantable surface of genus $g$. For $n\ge 14$ and $g\le 2\lfloor{n/2}\rfloor+1$ every such homomorphism is either cyclic, or it maps…

We show that surface groups are flexibly stable in permutations. This is the first non-trivial example of a non-amenable flexibly stable group. Our method is purely geometric and relies on an analysis of branched covers of hyperbolic…

We study monochromatic random waves on $\mathbb{R}^n$ defined by Gaussian variables whose variances tend to zero sufficiently fast. This has the effect that the Fourier transform of the monochromatic wave is an absolutely continuous measure…

An involution is a permutation that is its own inverse. Given a permutation $\sigma$ of $[n],$ let $\mathbf{N}_{n}(\sigma)$ denote the number of ways to write $\sigma$ as a product of two involutions of $[n].$ If we endow the symmetric…

Let $\Sigma $ be a compact connected and oriented surface with nonempty boundary and let $G$ be a Lie group equipped with a bi-invariant pseudo-Riemannian metric. The moduli space of flat principal $G$-bundles over $\Sigma$ which are…

We show that the expected gonality of a random graph is asymptotic to the number of vertices.

A rather natural construction for a smooth random surface in space is the level surface of value zero, or 'nodal' surface f(x,y,z)=0, of a (real) random function f; the interface between positive and negative regions of the function. A…

We consider random geometric graphs on the plane characterized by a non-uniform density of vertices. In particular, we introduce a graph model where $n$ vertices are independently distributed in the unit disc with positions, in polar…

We associate a noncommutative curve to a periodic, bipartite, planar dimer model with polygonal boundary. It determines the inverse Kasteleyn matrix and hence all correlations. It may be seen as a quantization of the limit shape…

We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems…

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family ($\mathrm{BD}_L$, $0 < L < \infty$) of random metric spaces homeomorphic to the closed…

The second author had previously obtained explicit generating functions for moments of characteristic polynomials of permutation matrices (n points). In this paper, we generalize many aspects of this situation. We introduce random shifts of…

Let B(n) be the set of pairs of permutations from the symmetric group of degree n with a 3-cycle commutator, and let A(n) be the set of those pairs which generate the symmetric or the alternating group of degree n. We find effective…

Let K be a d-dimensional convex body, and let $K^{(n)}$ be the intersection of n halfspaces containing $K$ whose bounding hyperplanes are independent and identically distributed. Under suitable distributional assumptions, we prove an…

A random algebraic graph is defined by a group $G$ with a uniform distribution over it and a connection $\sigma:G\longrightarrow[0,1]$ with expectation $p,$ satisfying $\sigma(g)=\sigma(g^{-1}).$ The random graph…