Related papers: Combinatorially random curves on surfaces
We study a natural model of random 2-dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$-face is included independently with probability p. Our main result is to exhibit a sharp…
We discuss the topological nature of the boundary spacetime, the conformal infinity of the ambient cosmological metric. Due to the existence of a homothetic group, the bounding spacetime must be equipped not with the usual Euclidean metric…
Consider any random graph model where potential edges appear independently, with possibly different probabilities, and assume that the minimum expected degree is omega(ln n). We prove that the adjacency matrix and the Laplacian of that…
Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in…
We study the topology of the space of probability measures invariant under the geodesic flow, defined on the unit-tangent bundle of a compact Riemannian manifold with non-positive curvature. Building on a previous work by Coud\`ene and…
We study properties of the Weyl pseudometric associated with an action of a countable amenable group on a compact metric space. We prove that the topological entropy and the number of minimal subsets of the closure of an orbit are both…
Consider a closed surface $M$ with negative Euler characteristic, and an admissible probability measure on the fundamental group of $M$ with finite first moment. Corresponding to each point in the Teichm\"uller space of $M$, there is an…
Fix an arbitrary compact orientable surface with a boundary and consider a uniform bipartite random quadrangulation of this surface with $n$ faces and boundary component lengths of order $\sqrt n$ or of lower order. Endow this…
Global physical properties of random media change qualitatively at a percolation threshold, where isolated clusters merge to form one infinite connected component. The precise knowledge of percolation thresholds is thus of paramount…
We investigate subsets of the symmetric group with structure similar to that of a graph. The trees of these subsets correspond to minimal conjugate generating sets of the symmetric group. There are two main theorems in this paper. The first…
We show that for any weakly convergent sequence of ergodic $SL_2(\mathbb{R})$-invariant probability measures on a stratum of unit-area translation surfaces, the corresponding Siegel-Veech constants converge to the Siegel-Veech constant of…
Let $G$ be a group and let $S$ be an inverse-closed and identity-free generating set of $G$. The \emph{Cayley graph} $\Cay(G,S)$ has vertex-set $G$ and two vertices $u$ and $v$ are adjacent if and only if $uv^{-1}\in S$. Let $CAY_d(n)$ be…
For a harmonic map $u:M^3\to S^1$ on a closed, oriented $3$--manifold, we establish the identity $$2\pi \int_{\theta\in S^1}\chi(\Sigma_{\theta})\geq \frac{1}{2}\int_{\theta\in S^1}\int_{\Sigma_{\theta}}(|du|^{-2}|Hess(u)|^2+R_M)$$ relating…
The goal is to show that an edge-reinforced random walk on a graph of bounded degree, with reinforcement weight function $W$ taken from a general class of reciprocally summable reinforcement weight functions, traverses a random attracting…
We investigate a class of line ensembles whose local structure is described by independent geometric random walk bridges, which have been conditioned to interlace with each other. The latter arise naturally in the context Schur processes,…
The dissertation is related to combinatorial geometry with a strong probabilistic flavor. The main results can be split into three parts. The results of the first part guarantee that each "unit distance graph" in the plane has an induced…
A pseudo-Anosov mapping class acts on Teichm\"uller space $\mathcal{T}$ as well as on the curve graph $\mathcal{C}$ with so called north-south dynamics. We can measure a stable translation length $l_\mathcal{T}$ and $l_\mathcal{C}$ of the…
The Dirichlet p-Laplacian in tubes of arbitrary cross-section along infinite curves in Euclidean spaces of arbitrary dimension is investigated. First, it is shown that the gap between the lowest point of the generalised spectrum and the…
We investigate oriented graphs based on the curve complex $C(S)$ of a closed surface $S$ and induced by functions on the vertex set of $C(S)$. In particular, we introduce the Dehn quasi-homothetic functions, which behave similarly to…
This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse…