Symmetry breaking in two-dimensional square grids: persistence and failure of the dimensional crossover

TL;DR

This paper investigates how defects in infinite 2D square grids influence the persistence or failure of a dimensional crossover phenomenon related to Sobolev inequalities and nonlinear Schrödinger ground states.

Contribution

It analyzes the robustness of the dimensional crossover in square grids under various topological defects, identifying conditions for preservation or failure.

Findings

Certain defects preserve the dimensional crossover.

Other defects cause the crossover to fail.

The study clarifies the topological conditions affecting the phenomenon.

Abstract

We discuss the model robustness of the infinite two-dimensional square grid with respect to symmetry breakings due to the presence of defects, that is, lacks of finitely or infinitely many edges. Precisely, we study how these topological perturbations of the square grid affect the so-called dimensional crossover identified in [Adami et al. 2019]. Such a phenomenon has two evidences: the coexistence of the one and the two-dimensional Sobolev inequalities and the appearence of a continuum of $L^2$-critical exponents for the ground states at fixed mass of the nonlinear Schr\"odinger equation. From this twofold perspective, we investigate which classes of defects do preserve the dimensional crossover and which classes do not.