p-Groups acting on Riemann surfaces

Let p be a prime integer and let $r\ge 3$ be an integer so that $p\ge 5r?7$. We show that a closed Riemann surface S of genus $g\ge 2$ has at most one p-group H of conformal automorphisms so that $S/H$ has genus zero and exactly r cone points. This, in particular, asserts that, for $r=3$ and $p\ge 11$, the minimal field of definition of S coincides with that of $(S,H)$. Another application of this fact, for the case that S is pseudo-real, is that $\mathrm{Aut}(S)/H$ must be either trivial or a cyclic group and that r is necessarily even. This generalizes a result due to Bujalance–Costa for the case of pseudo-real cyclic p-gonal Riemann surfaces.     

Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices

In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from $n$ independent observations of a $p$-dimensional time series with iid components converge almost surely to ${(1+\sqrt{\gamma })}^2$ and ${(1?\sqrt{\gamma })}^2$, respectively, as $n\to \infty$, if $p∕n\to \gamma \in (0,1]$ and the truncated variance of the entry distribution is “almost slowly varying”, a condition we describe via moment properties of self-normalized sums. Moreover, the empirical spectral distributions of these sample correlation matrices converge weakly, with probability $1$, to the Mar?enko–Pastur law, which extends a result in Bai and Zhou (2008). We compare the behavior of the eigenvalues of the sample covariance and sample correlation matrices and argue that the latter seems more robust, in particular in the case of infinite fourth moment. We briefly address some practical issues for the estimation of extreme eigenvalues in a simulation study.In our proofs we use the method of moments combined with a Path-Shortening Algorithm, which efficiently uses the structure of sample correlation matrices, to calculate precise bounds for matrix norms. We believe that this new approach could be of further use in random matrix theory.     

Supercritical loop percolation on Zd for d≥3

We consider supercritical percolation on ${\mathbb{Z}}^d$ ($d\ge 3$) induced by random walk loop soup. Two vertices are in the same cluster if they are connected through a sequence of intersecting loops. We obtain quenched parabolic Harnack inequalities, Gaussian heat kernel bounds, the invariance principle and the local central limit theorem for the simple random walks on the unique infinite cluster. We also show that the diameter of finite clusters have exponential tails like in Bernoulli bond percolation. Our results hold for all $d\ge 3$ and all supercritical intensities despite polynomial decay of correlations.     

Existence of unimodular elements in a projective module

(1) Let R be an affine algebra over an algebraically closed field of characteristic 0 with $\dim (R)=n$. Let P be a projective $A=R[T_1,?,T_k]$-module of rank n with determinant L. Suppose I is an ideal of A of height n such that there are two surjections $\alpha :P?I$ and $?:L\oplus A^{n?1}?I$. Assume that either (a) $k=1$ and $n\ge 3$ or (b) k is arbitrary but $n\ge 4$ is even. Then P has a unimodular element (see 4.1, 4.3).(2) Let R be a ring containing $\mathbb{Q}$ of even dimension n with height of the Jacobson radical of $R\ge 2$. Let P be a projective $R[T,T^{?1}]$-module of rank n with trivial determinant. Assume that there exists a surjection $\alpha :P?I$, where $I?R[T,T^{?1}]$ is an ideal of height n such that I is generated by n elements. Then P has a unimodular element (see 3.4).     

Generalized Brjuno functions associated to α-continued fractions

For $0\le \alpha \le 1$ given, we consider the one-parameter family of $\alpha$-continued fraction maps, which include the Gauss map ($\alpha =1$), the nearest integer ($\alpha =1/2$) and by-excess ($\alpha =0$) continued fraction maps. To each of these expansions and to each choice of a positive function $u$ on the interval ${\mathbb{I}}_{\alpha }$ we associate a generalized Brjuno function $B_{(\alpha ,u)}\left(x\right)$. When $\alpha =1/2$ or $\alpha =1$, and $u\left(x\right)=?log\left(x\right)$, these functions were introduced by Yoccoz in his work on linearization of holomorphic maps.We compare the functions obtained with different values of $\alpha$ and we prove that the set of $(\alpha ,u)$-Brjuno numbers does not depend on the choice of $\alpha$ provided that $\alpha \ne 0$. We then consider the case $\alpha =0$, $u\left(x\right)=?log\left(x\right)$ and we prove that $x$ is a Brjuno number (for $\alpha \ne 0$) if and only if both $x$ and $?x$ are Brjuno numbers for $\alpha =0$.     

Systems of stochastic Poisson equations: Hitting probabilities

We consider a $d$-dimensional random field $u=(u\left(x\right),x\in D)$ that solves a system of elliptic stochastic equations on a bounded domain $D?{\mathbb{R}}^k$, with additive white noise and spatial dimension $k=1,2,3$. Properties of $u$ and its probability law are proved. For Gaussian solutions, using results from Dalang and Sanz-Solé (2009), we establish upper and lower bounds on hitting probabilities in terms of the Hausdorff measure and Bessel–Riesz capacity, respectively. This relies on precise estimates of the canonical distance of the process or, equivalently, on $L^2$ estimates of increments of the Green function of the Laplace equation.     

Improved degree conditions for Hamiltonian properties

In 1980, Bondy proved that for an integer $k\ge 2$ a $(k+s)$-connected graph of order $n\ge 3$ is traceable ($s=?1$) or Hamiltonian ($s=0$) or Hamiltonian-connected ($s=1$) if the degree sum of every set of $k+1$ pairwise nonadjacent vertices is at least $\frac{1}{2}((k+1)(n+s?1)+1)$. This generalizes the well-known sufficient conditions of Dirac ($k=0$) and Ore ($k=1$). The condition in Bondy’s Theorem is not tight for $k\ge 2$. We improve this sufficient degree condition and show the general tightness of this result.     

Equivalent martingale measures for Lévy-driven moving averages and related processes

In the present paper we obtain sufficient conditions for the existence of equivalent local martingale measures for Lévy-driven moving averages and other non-Markovian jump processes. The conditions that we obtain are, under mild assumptions, also necessary. For instance, this is the case for moving averages driven by an $\alpha$-stable Lévy process with $\alpha \in (1,2]$.Our proofs rely on various techniques for showing the martingale property of stochastic exponentials.     

Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in (1/2,1)

This paper addresses the exponential stability of the trivial solution of some types of evolution equations driven by Hölder continuous functions with Hölder index greater than 1/2. The results can be applied to the case of equations whose noisy inputs are given by a fractional Brownian motion $B^H$ with covariance operator Q, provided that $H\in (1/2,1)$ and $\mathrm{tr}(Q)$ is sufficiently small.     

On vertex-disjoint cycles and degree sum conditions

This paper considers a degree sum condition sufficient to imply the existence of $k$ vertex-disjoint cycles in a graph $G$. For an integer $t\ge 1$, let ${\sigma }_t\left(G\right)$ be the smallest sum of degrees of $t$ independent vertices of $G$. We prove that if $G$ has order at least $7k+1$ and ${\sigma }_4\left(G\right)\ge 8k?3$, with $k\ge 2$, then $G$ contains $k$ vertex-disjoint cycles. We also show that the degree sum condition on ${\sigma }_4\left(G\right)$ is sharp and conjecture a degree sum condition on ${\sigma }_t\left(G\right)$ sufficient to imply $G$ contains $k$ vertex-disjoint cycles for $k\ge 2$.     

Equation-regular sets and the Fox–Kleitman conjecture

Given $k\ge 1$, the Fox–Kleitman conjecture from 2006 states that there exists a nonzero integer $b$ such that the $2k$-variable linear Diophantine equation

$\sum _{i=1}^k(x_i?y_i)=b$

is $(2k?1)$-regular. This is best possible, since Fox and Kleitman showed that for all $b\ge 1$, this equation is not $2k$-regular. While the conjecture has recently been settled for all $k\ge 2$, here we focus on the case $k=3$ and determine the degree of regularity of the corresponding equation for all $b\ge 1$. In particular, this independently confirms the conjecture for $k=3$. We also briefly discuss the case $k=4$.     

Descendant sets in infinite,primitive highly arc transitive digraphs with prime power out-valency

The descendant set $\mathrm{desc}\left(\alpha \right)$ of a vertex $\alpha$ in a directed graph (digraph) is the subdigraph on the set of vertices reachable by a directed path from $\alpha$. We study the structure of descendant sets $\Gamma$ in an infinite, primitive, highly arc transitive digraph with out-valency $p^k$, where $p$ is a prime and $k\ge 1$. It was already known that $\Gamma$ is a tree when $k=1$ and we show the same holds when $k=2$. However, for $k\ge 3$ there are examples of infinite, primitive highly arc transitive digraphs of out-valency $p^k$ whose descendant sets are not trees, for some prime $p$.