Improved Replica Bounds for the Independence Ratio of Random Regular Graphs

Abstract

AbstractStudying independent sets of maximum size is equivalent to considering the hard-core model with the fugacity parameter $$\lambda $$ λ 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 $$d \ge 20$$ d ≥ 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 $$d \ge 3$$ d ≥ 3 . For $$d \le 19$$ d ≤ 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 ($$r=2,3,4,5$$ r = 2 , 3 , 4 , 5 ) to obtain improved rigorous upper bounds for the independence ratio for each degree $$3 \le d \le 19$$ 3 ≤ d ≤ 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.