Boosted second moment method in random regular graphs
Balázs Gerencsér, Viktor Harangi
TL;DR
This work advances explicit lower bounds for the asymptotic independence ratio $\alpha^*_d$ of random $d$-regular graphs by refining the second-moment method with a spatial Markov property, enabling local augmentations of independent sets and yielding improved bounds for $d\ge 10$. The authors formalize a framework of typical and invariant processes on the infinite $d$-regular tree $T_d$, leveraging entropy $\Sigma(\pi)$ and a strengthened second-moment condition to guarantee the existence of Markovian independent sets with prescribed densities. They introduce augmentations via full-zero vertices to derive higher-density independent sets, with explicit expressions $\hat{\alpha}=\alpha+(1-\alpha)p^d\frac{1+(1-p^{d-1})^d}{2}$, and demonstrate applicability to star-decomposition problems by thinning to obtain $k$-star decompositions for certain $d,k$. The results come with computational tools (Sage code and a website) enabling explicit bounds for large ranges of $d$, reinforcing the potential of this approach to close gaps with conjectured 1-RSB predictions and related bounds. Overall, the paper presents a versatile, close-to-optimal methodology for proving richer structural properties in random regular graphs beyond independent sets, including decompositions into stars.
Abstract
Determining the asymptotic independence ratio of random regular graphs is a key challenge in the area of sparse random graphs. Due to the interpolation method, very good upper bounds are known, which are actually known to be sharp for sufficiently large degrees. This paper provides explicit lower bounds for given degrees $d$, beating the previous best bounds for $d \geq 10$. The starting point is a second moment argument that we can boost by arguing that the obtained independent set has a certain spatial Markov property. One can then exploit this property by making local modifications to the independent set, resulting in substantial improvements. Our approach can also be used to prove the existence of other objects in random regular graphs. To demonstrate this, we consider the problem of decomposing random regular graphs into stars.