Related papers: Some results on random unimodular lattices

In [A. Stolz and A. Thom, On the lattice of normal subgroups in ultraproducts of compact simple groups, PLMS 108(1), 2014] it was stated that the lattice of normal subgroups of an ultraproduct of finite simple groups is always linearly…

We prove that the lattice of normal subgroups of ultraproducts of compact simple non-abelian groups is distributive. In the case of ultraproducts of finite simple groups or compact connected simple Lie groups of bounded rank the set of…

A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice $\mathbb{Z}^{2}$, gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak…

We investigate predicative aspects of order theory in constructive univalent foundations. By predicative and constructive, we respectively mean that we do not assume Voevodsky's propositional resizing axioms or excluded middle. Our work…

We consider Euclidean lattices spanned by images of algebraic conjugates of an algebraic number under Minkowski embedding, investigating their rank, properties of their automorphism groups and sets of minimal vectors. We are especially…

It is a result of Ginibre that the normalized bulk $k$-point correlation functions of a complex $n\times n$ Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process…

It is well known that the orbit of a lattice in hyperbolic $n$-space is uniformly distributed when projected radially onto the unit sphere. In the present work, we consider the fine-scale statistics of the projected lattice points, and…

A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of ${\mathbb Z}^2$, as well as their determinant and minima sets. We…

For a random matrix of entries sampled independently from a fairly general distribution in Z we study the probability that the cokernel is isomorphic to a given finite abelian group, or when it is cyclic. This includes the probability that…

In this paper we determine, under some mild restrictions, the lattice of submodules $\gL$ of a module $M$ all of whose composition factors have multiplicity one. Such a lattice is distributive, and hence determined by its poset of down-sets…

Mixtures of $r$ independent distributions for two discrete random variables can be represented by matrices of nonnegative rank $r$. Likelihood inference for the model of such joint distributions leads to problems in real algebraic geometry…

Let $\Lambda = \mathrm{SL}_2(\Bbb Z)$ be the modular group and let $c_n(\Lambda)$ be the number of congruence subgroups of $\Lambda$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log c_n(\Lambda)}{(\log n)^2/\log\log…

Let $L$ denote a finite lattice with at least two points and let $A$ denote the incidence algebra of $L$. We prove that $L$ is distributive if and only if $A$ is an Auslander regular ring, which gives a homological characterisation of…

Consider $n$ $d$-dimensional vectors with iid entries from a lattice distribution $X$. We show that the probability that all distances between them are equal is asymptotically \[ C_n\cdot\frac{1}{d^{(m-1)/2}} \quad \text{for} \quad d \to…

We prove the universal asymptotically almost sure non-singularity of general Ginibre and Wigner ensembles of random matrices when the distribution of the entries are independent but not necessarily identically distributed and may depend on…

In this paper we develop a theory of convexity for a free Abelian group M (the lattice of integer points), which we call theory of discrete convexity. We characterize those subsets X of the group M that could be call "convex". One property…

We show that any lattice in $\mathrm{SL}_3(k)$, where $k$ is a nonarchimedean local field, contains an undistorted subgroup isomorphic to the free product $\mathbb{Z}^2*\mathbb{Z}$. To our knowledge, the subgroups we construct give the…

We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\mathbb{Z}^2$. We first prove a formula on the rotation number of a…

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank $3$): (a) The number of extreme points in an…

We are interested in the study of Gibbs and equilbrium probabilities on the lattice $\mathbb{R}^{\mathbb{N}}$. Consider the unilateral full-shift defined on the non-compact set $\mathbb{R}^{\mathbb{N}}$ and an $\alpha$-H\"older continuous…