The Kannan-Tetali-Vempala conjecture for simple undirected graphs

Claims Theorem 1.1: for every $n\ge4$ and every graphical labeled degree vector $d$, the lazy switch chain has total-variation mixing time at most $2n^8$ and, if $|\Omega_d|>1$, spectral gap at least $[24n^2\binom n4]^{-1}$. Corollary: an exactly uniform sampler for any graphical $d$ with expected polynomial bit running time, terminating almost surely. This settles the simple undirected formulation; the bipartite formulation was proved earlier by Fu, Qin and Wang (listed separately), and the directed formulation is not addressed. The exponent 8 is not claimed to be sharp.

Claims Theorem 1.1: for every $n\ge4$ and every graphical labeled degree vector $d$, the lazy switch chain has total-variation mixing time at most $2n^8$ and, if $|\Omega_d|>1$, spectral gap at least $[24n^2\binom n4]^{-1}$. Corollary: an exactly uniform sampler for any graphical $d$ with expected polynomial bit running time, terminating almost surely. This settles the simple undirected formulation; the bipartite formulation was proved earlier by Fu, Qin and Wang (listed separately), and the directed formulation is not addressed. The exponent 8 is not claimed to be sharp.