Random Latin squares and 2-dimensional expanders

TL;DR

This paper introduces a new random model for 2-dimensional complexes and proves the existence of infinite families of expanders with bounded degree, advancing understanding of high-dimensional expanders.

Contribution

The paper presents a novel random model for 2-complexes and demonstrates the existence of infinite families of 2-dimensional expanders with bounded maximum edge degree.

Findings

Existence of infinite families of 2D -expanders

Construction of expanders with fixed  and degree d

New probabilistic model for random 2-complexes

Abstract

Let X be a 2-dimensional simplicial complex. The degree of an edge e is the number of 2-faces of X containing e. The complex X is an \epsilon-expander if the coboundary d_1(\phi) of every Z_2-valued 1-cochain \phi \in C^1(X;Z_2) satisfies |support(d_1(\phi))| \geq \epsilon |\supp(\phi+d_0(\psi))| for some 0-cochain \psi. Using a new model of random 2-complexes we show the existence of an infinite family of 2-dimensional \epsilon-expanders with maximum edge degree d, for some fixed \epsilon>0 and d.