ferromagnetic: Byy >1 FPRAS:[Jerrum-Sinclair'93][Goldberg-Jerrum-Paterson'03] anti-ferromagnetic: By<1 hardcore model:B=0,y =1 [Weitz'06] Ising model:B=y [Sinclair-Srivastava-Thurley'12] (B,y,A)lies in the interior of FPTAS for graphs uniqueness region of A-regular tree of max-degree△ 2.5 -By=1 uiqueness threshold threshold achieved by heatbath random walk [Goldberg-Jerrum-Paterson'03] 15 [Li-Lu-Y.'12]: 0.5 0<B.y<1 FPTAS for arbitrary graphs 0 0.5 1.5 2.5ferromagnetic: [Jerrum-Sinclair’93] ￾￾ > 1 FPRAS: [Goldberg-Jerrum-Paterson’03] anti-ferromagnetic: ￾￾ < 1 hardcore model: Ising model: ￾ = 0, ￾ = 1 ￾ = ￾ ∃ FPTAS for graphs of max-degree Δ (β, γ, λ) lies in the interior of uniqueness region of Δ-regular tree [Sinclair-Srivastava-Thurley’12] [Weitz’06] 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
0< , <1 = 1 uniqueness threshold threshold achieved by heatbath random walk ￾ ￾ [Li-Lu-Y. ’12]: FPTAS for arbitrary graphs [Goldberg-Jerrum-Paterson’03]