We present a new lower bound on the spectral gap of the Glauber dynamics for the Gibbs distribution of a spectrally independent $q$-spin system . Previously, such a result was known either if the girth of $G$ is at least $5$ [Dyer et.~al, FOCS 2004], or underrestrictions on $Delta$ . We show that for a regular graph $G = (V,E)$ with degree $Delta \geq 3$ with girth $6$ and for any $Delta > 0$ partitionfunction of the hardcore model may be approximated within a$(1+\varepsilon)$-multiplicative factor in time$\tilde{O}_{\delta}(n = O(1)$ Withhigh probability, an approximately uniformly random matching may be sampled intime $O(1). This improves the corresponding running time of$O_d(n,d/n) The new bound covers the entire regime of $D/n/n, d/n is$tilde(1-\d delta)

Author(s) : Vishesh Jain, Huy Tuan Pham, Thuy Duong Vuong