Improved replica bounds for the independence ratio of random regular graphs

TL;DR

This paper develops improved upper bounds for the independence ratio of random regular graphs with degrees 3 to 19 by numerically optimizing discrete RSB formulas, advancing understanding in statistical physics and graph theory.

Contribution

It introduces numerical optimization of discrete RSB formulas to obtain tighter bounds for the independence ratio in degrees 3 to 19.

Findings

Improved upper bounds for degrees 3 to 19.

Use of numerical optimization for complex RSB formulas.

Demonstrated challenges in global optimization due to local minima.

Abstract

Studying independent sets of maximum size is equivalent to considering the hard-core model with the fugacity parameter $\lambda$ tending to infinity. Finding the independence ratio of random $d$-regular graphs for some fixed degree $d$ has received much attention both in random graph theory and in statistical physics. For $d \geq 20$ the problem is conjectured to exhibit 1-step replica symmetry breaking (1-RSB). The corresponding 1-RSB formula for the independence ratio was confirmed for (very) large $d$ in a breakthrough paper by Ding, Sly, and Sun. Furthermore, the so-called interpolation method shows that this 1-RSB formula is an upper bound for each $d \geq 3$. For $d \leq 19$ this bound is not tight and full-RSB is expected. In this work we use numerical optimization to find good substituting parameters for discrete $r$-RSB formulas ($r=2,3,4,5$) to obtain improved rigorous upper bounds for the independence ratio for each degree $3 \leq d \leq 19$. As $r$ grows, these formulas get increasingly complicated and it becomes challenging to compute their numerical values efficiently. Also, the functions to minimize have a large number of local minima, making global optimization a difficult task.