sampling(with bounded error)from the hardcore model on graphs with max-degree△and fugacityλ>0 uniqueness threshold:(=(A-1)(4-1) e ≈ (△-2)A △-2 [Glauber'63]discovery of Glauber dynamic [Luby-Vigoda'99][Dyer-Greenhill'00][Vigoda'01] tmix=O(log n))for Glauber dynamics when入<2/（△-2） ·[Weitz'06]poly-time sampler whenλ<λc(△)for△=O(l)
• [Glauber’63] discovery of Glauber dynamic • [Luby-Vigoda’99][Dyer-Greenhill’00][Vigoda’01] τmix=O(nlog n) for Glauber dynamics when λ<2/(Δ-2) • [Weitz’06] poly-time sampler when λ<λc(Δ) for Δ=O(1) sampling (with bounded error) from the hardcore model on graphs with max-degree Δ and fugacity λ>0 c() = ( 1)(1) ( 2) ⇡ e 2 uniqueness threshold:
sampling(with bounded error)from the hardcore model on graphs with max-degree△and fugacityλ>0 uniqueness threshold:(=(A-1)(-1) e (△-2)A ≈△-2 [Glauber'63]discovery of Glauber dynamic [Luby-Vigoda'99][Dyer-Greenhill'00][Vigoda'01] tmix=O(nlog n)for Glauber dynamics whenλ<2/（△-2） O(1og△)-tim9 ●[Weitz'06]oy-time sampler whenλ<入c(△)for△=O(l)
• [Glauber’63] discovery of Glauber dynamic • [Luby-Vigoda’99][Dyer-Greenhill’00][Vigoda’01] τmix=O(nlog n) for Glauber dynamics when λ<2/(Δ-2) • [Weitz’06] poly-time sampler when λ<λc(Δ) for Δ=O(1) sampling (with bounded error) from the hardcore model on graphs with max-degree Δ and fugacity λ>0 c() = ( 1)(1) ( 2) ⇡ e 2 uniqueness threshold: nO(log )-time
sampling(with bounded error)from the hardcore model on graphs with max-degree△and fugacityλ>0 uniqueness threshold:A.(△)=A-l)a-” e (△-2)A △-2 [Glauber'63]discovery of Glauber dynamic [Luby-Vigoda'99][Dyer-Greenhill'00][Vigoda'01] tmix=O(log)for Glauber dynamics when入<2/（△-2） O(log△)-tim9。 m ·[Weitz'06]oy-time sampler whenλ<λc(A)for△=O(l) [Mossel-Weitz-Wormald'09]slow mixing of Glauber dynamics when>入c(△)
• [Glauber’63] discovery of Glauber dynamic • [Luby-Vigoda’99][Dyer-Greenhill’00][Vigoda’01] τmix=O(nlog n) for Glauber dynamics when λ<2/(Δ-2) • [Weitz’06] poly-time sampler when λ<λc(Δ) for Δ=O(1) • [Mossel-Weitz-Wormald’09] slow mixing of Glauber dynamics when λ>λc(Δ) sampling (with bounded error) from the hardcore model on graphs with max-degree Δ and fugacity λ>0 c() = ( 1)(1) ( 2) ⇡ e 2 uniqueness threshold: nO(log )-time
sampling (with bounded error)from the hardcore model on graphs with max-degree△and fugacityλ>0 uniqueness threshold:()=(A-1)(4-1) e ≈ (△-2)A △-2 [Glauber'63]discovery of Glauber dynamic [Luby-Vigoda'99][Dyer-Greenhill'00][Vigoda'01] tmix=O(nlog n)for Glauber dynamics when入<2/（△-2） nO(1og△)-time。 ·[Weitz'06]oiy-time sampler when<λc(△)for△=O(1) ● [Mossel-Weitz-Wormald'09]slow mixing of Glauber dynamics when>λc(△) ·[Sly'lO]NP-hardness of sampling when>λc(△)
• [Glauber’63] discovery of Glauber dynamic • [Luby-Vigoda’99][Dyer-Greenhill’00][Vigoda’01] τmix=O(nlog n) for Glauber dynamics when λ<2/(Δ-2) • [Weitz’06] poly-time sampler when λ<λc(Δ) for Δ=O(1) • [Mossel-Weitz-Wormald’09] slow mixing of Glauber dynamics when λ>λc(Δ) • [Sly’10] NP-hardness of sampling when λ>λc(Δ) sampling (with bounded error) from the hardcore model on graphs with max-degree Δ and fugacity λ>0 c() = ( 1)(1) ( 2) ⇡ e 2 uniqueness threshold: nO(log )-time
sampling(with bounded error)from the hardcore model on graphs with max-degree A and fugacity A>0 uniqueness threshold:()=(A-1)(-1) e ≈ (△-2)A △-2 [Glauber'63]discovery of Glauber dynamic [Luby-Vigoda'99][Dyer-Greenhill'00][Vigoda'01] Tmix=O(nlog n)for Glauber dynamics when <2/(A-2) O(1og△)-time。 m ● [Weitz'06]oiy-time sampler whenλ<入c(△)for△=O(1) ● [Mossel-Weitz-Wormald'09]slow mixing of Glauber dynamics when>λc(△) [Slyl0]NP-hardness of sampling when>λc(△) Theorem (Efthymiou-Hayes-Stefankovic-Vigoda-Y.'16): tmix=O(logn)for Glauber dynamics whenλ<c(△)， if A is sufficiently large,and there is no small cycles
• [Glauber’63] discovery of Glauber dynamic • [Luby-Vigoda’99][Dyer-Greenhill’00][Vigoda’01] τmix=O(nlog n) for Glauber dynamics when λ<2/(Δ-2) • [Weitz’06] poly-time sampler when λ<λc(Δ) for Δ=O(1) • [Mossel-Weitz-Wormald’09] slow mixing of Glauber dynamics when λ>λc(Δ) • [Sly’10] NP-hardness of sampling when λ>λc(Δ) sampling (with bounded error) from the hardcore model on graphs with max-degree Δ and fugacity λ>0 c() = ( 1)(1) ( 2) ⇡ e 2 uniqueness threshold: nO(log )-time Theorem (Efthymiou-Hayes-Štefankovič-Vigoda-Y.’16): τmix=O(nlog n) for Glauber dynamics when λ<λc(Δ), if Δ is sufficiently large, and there is no small cycles