Spatial Mixing and the Connective Constant: optimal bounds Yitong - PowerPoint PPT Presentation

Spatial Mixing and the Connective Constant: optimal bounds Yitong Yin Nanjing University Alistair Sinclair ( UC Berkeley ) Joint work with: Piyush Srivastava ( UC Berkeley ) Daniel Š tefankovi č ( Rochester )

undirected graph G = ( V, E ) matchings { } approximately counting # of of G independent sets matching { } almost uniformly sampling a of G independent set

undirected graph G = ( V, E ) matchings { } approximately counting # of of G independent sets computationally equivalent matching { } almost uniformly sampling a of G independent set

undirected graph G = ( V, E ) monomer-dimer model: set of all matchings M ( G ) partition function X λ | M | Z λ ( G ) = M ∈ M ( G )

undirected graph G = ( V, E ) monomer-dimer model: set of all matchings M ( G ) partition function X λ | M | Z λ ( G ) = M ∈ M ( G ) λ | M | Gibbs distribution µ ( M ) = Z λ ( G )

undirected graph G = ( V, E ) monomer-dimer model: set of all matchings M ( G ) partition function X λ | M | Z λ ( G ) = M ∈ M ( G ) λ | M | Gibbs distribution µ ( M ) = Z λ ( G ) hardcore model: set of all independent sets I ( G ) partition function X λ | I | Z λ ( G ) = I ∈ I ( G ) λ | I | Gibbs distribution µ ( I ) = Z λ ( G )

Known Results computing the partition function

Known Results computing the partition function • monomer-dimer (matching)

Known Results computing the partition function • monomer-dimer (matching) • FPRAS by MCMC [Jerrum, Sinclair 1989]

Known Results computing the partition function • monomer-dimer (matching) • FPRAS by MCMC [Jerrum, Sinclair 1989] • FPTAS for graphs with bounded max-degree [Bayati et al , STOC 2007]

Known Results computing the partition function • monomer-dimer (matching) • FPRAS by MCMC [Jerrum, Sinclair 1989] • FPTAS for graphs with bounded max-degree [Bayati et al , STOC 2007] • hardcore (independent set)

Known Results computing the partition function • monomer-dimer (matching) • FPRAS by MCMC [Jerrum, Sinclair 1989] • FPTAS for graphs with bounded max-degree [Bayati et al , STOC 2007] • hardcore (independent set) • FPRAS for counting independent sets ( λ =1 ) of graphs with max-degree ≤ 4 [Dyer, Greenhill 2000; Vigoda 2001]

Known Results computing the partition function • monomer-dimer (matching) • FPRAS by MCMC [Jerrum, Sinclair 1989] • FPTAS for graphs with bounded max-degree [Bayati et al , STOC 2007] • hardcore (independent set) • FPRAS for counting independent sets ( λ =1 ) of graphs with max-degree ≤ 4 [Dyer, Greenhill 2000; Vigoda 2001] • FPTAS for graphs with max-degree d when λ < λ c ( d -1) [Weitz, STOC 2006]

Known Results computing the partition function • monomer-dimer (matching) • FPRAS by MCMC [Jerrum, Sinclair 1989] • FPTAS for graphs with bounded max-degree [Bayati et al , STOC 2007] • hardcore (independent set) • FPRAS for counting independent sets ( λ =1 ) of graphs with max-degree ≤ 4 [Dyer, Greenhill 2000; Vigoda 2001] • FPTAS for graphs with max-degree d when λ < λ c ( d -1) d d [Weitz, STOC 2006] uniqueness threshold: λ c ( d ) = ( d − 1) d +1

Known Results computing the partition function • monomer-dimer (matching) • FPRAS by MCMC [Jerrum, Sinclair 1989] • FPTAS for graphs with bounded max-degree [Bayati et al , STOC 2007] • hardcore (independent set) • FPRAS for counting independent sets ( λ =1 ) of graphs with max-degree ≤ 4 [Dyer, Greenhill 2000; Vigoda 2001] • FPTAS for graphs with max-degree d when λ < λ c ( d -1) d d [Weitz, STOC 2006] uniqueness threshold: λ c ( d ) = ( d − 1) d +1 • inapproxamable if λ > λ c ( d -1) [Sly, FOCS 2010; Sly, Sun, FOCS 2012]

Known Results computing the partition function • monomer-dimer (matching) • FPRAS by MCMC [Jerrum, Sinclair 1989] • FPTAS for graphs with bounded max-degree [Bayati et al , STOC 2007] • hardcore (independent set) spatial • mixing FPRAS for counting independent sets ( λ =1 ) of graphs with max-degree ≤ 4 [Dyer, Greenhill 2000; Vigoda 2001] • FPTAS for graphs with max-degree d when λ < λ c ( d -1) d d [Weitz, STOC 2006] uniqueness threshold: λ c ( d ) = ( d − 1) d +1 • inapproxamable if λ > λ c ( d -1) [Sly, FOCS 2010; Sly, Sun, FOCS 2012]

Connective Constants [Madras, Slade 1996] N ( v, ` ) : number of paths of length l starting from v

Connective Constants [Madras, Slade 1996] N ( v, ` ) : number of paths of length l starting from v for an infinite graph G : connective constant N ( v, ` ) 1 / ` ∆ ( G ) = sup lim sup v ∈ V ` →∞

Connective Constants [Madras, Slade 1996] N ( v, ` ) : number of paths of length l starting from v for an infinite graph G : connective constant N ( v, ` ) 1 / ` ∆ ( G ) = sup lim sup v ∈ V ` →∞ can be similarly defined for a family of finite graphs G

Connective Constants [Madras, Slade 1996] N ( v, ` ) : number of paths of length l starting from v for an infinite graph G : connective constant N ( v, ` ) 1 / ` ∆ ( G ) = sup lim sup v ∈ V ` →∞ can be similarly defined for a family of finite graphs G the connective constant represents the average growth rate of number of paths from a vertex

Connective Constants [Madras, Slade 1996] N ( v, ` ) : number of paths of length l starting from v for an infinite graph G : connective constant N ( v, ` ) 1 / ` ∆ ( G ) = sup lim sup v ∈ V ` →∞ can be similarly defined for a family of finite graphs G the connective constant represents the average growth rate of number of paths from a vertex q for honeycomb lattice √ 2 + 2 [Duminil-Copin, Smirnov, Annals of Math 2012]

Our Results • monomer-dimer (matching): • FPTAS for graphs with bounded max-degree [Bayati et al , STOC 2007] • FPTAS for graphs with bounded connective constant. • hardcore (independent set): • FPTAS for graphs with max-degree d when λ < λ c ( d -1) [Weitz, STOC 2006] • FPTAS for graphs with connective constant Δ when λ < λ c ( Δ ) . ∆ ∆ uniqueness threshold: λ c ( ∆ ) = ( ∆ − 1) ∆ +1

monomer-dimer hardcore partition functions: X λ | M | X λ | I | Z λ ( G ) = Z λ ( G ) = M ∈ M ( G ) I ∈ I ( G ) λ | M | λ | I | Gibbs distributions: µ ( M ) = µ ( I ) = Z λ ( G ) Z λ ( G ) marginal Pr[ v is matched | σ Λ ] Pr[ v is occupied | σ Λ ] probabilities: σ Λ : configuration of being matched/unmatched or occupied/unoccupied for vertices in Λ ⊂ V

monomer-dimer hardcore partition functions: X λ | M | X λ | I | Z λ ( G ) = Z λ ( G ) = M ∈ M ( G ) I ∈ I ( G ) λ | M | λ | I | Gibbs distributions: µ ( M ) = µ ( I ) = Z λ ( G ) Z λ ( G ) marginal Pr[ v is matched | σ Λ ] Pr[ v is occupied | σ Λ ] probabilities: σ Λ : configuration of being matched/unmatched or occupied/unoccupied for vertices in Λ ⊂ V by self-reduction: efficient approximation of marginal probabilities implies efficient approximation of partition function (Jerrum-Valiant-Vazirani)

Spatial Mixing (Decay of Correlation) weak spatial mixing (WSM): Pr[ c ( v ) = x | σ ∂ R ] ≈ Pr[ c ( v ) = x | τ ∂ R ] error < exp (- t ) G ∂ R R t v

Spatial Mixing (Decay of Correlation) weak spatial mixing (WSM): Pr[ c ( v ) = x | σ ∂ R ] ≈ Pr[ c ( v ) = x | τ ∂ R ] error < exp (- t ) uniqueness threshold: WSM in d -regular tree G ∂ R R t v

Spatial Mixing (Decay of Correlation) weak spatial mixing (WSM): Pr[ c ( v ) = x | σ ∂ R ] ≈ Pr[ c ( v ) = x | τ ∂ R ] error < exp (- t ) G ∂ R R t v

Spatial Mixing (Decay of Correlation) weak spatial mixing (WSM): Pr[ c ( v ) = x | σ ∂ R ] ≈ Pr[ c ( v ) = x | τ ∂ R ] error < exp (- t ) strong spatial mixing (SSM): Pr[ c ( v ) = x | σ ∂ R , σ Λ ] ≈ Pr[ c ( v ) = x | τ ∂ R , σ Λ ] G ∂ R R t v Λ

Spatial Mixing (Decay of Correlation) weak spatial mixing (WSM): Pr[ c ( v ) = x | σ ∂ R ] ≈ Pr[ c ( v ) = x | τ ∂ R ] error < exp (- t ) strong spatial mixing (SSM): Pr[ c ( v ) = x | σ ∂ R , σ Λ ] ≈ Pr[ c ( v ) = x | τ ∂ R , σ Λ ] G SSM: the value of ∂ R Pr[ c ( v ) = x | σ Λ ] R t is approximable v Λ by local information

Self-Avoiding Walk Tree (Godsil 1981) G =( V , E ) v 1 4 3 2 5 6

Self-Avoiding Walk Tree (Godsil 1981) G =( V , E ) 1 v 1 4 3 2 5 6

Self-Avoiding Walk Tree (Godsil 1981) G =( V , E ) T = T ��� ( G, v ) 1 v 1 4 2 3 4 3 2 6 4 3 5 5 6 5 6 5 6 5 6

Self-Avoiding Walk Tree (Godsil 1981) G =( V , E ) T = T ��� ( G, v ) 1 v 1 4 2 3 4 3 2 6 4 3 5 σ Λ 5 6 6 5 6 5 6 5 6

Self-Avoiding Walk Tree (Godsil 1981) G =( V , E ) T = T ��� ( G, v ) 1 v 1 4 2 3 4 3 2 6 6 4 3 5 σ Λ 5 6 6 5 6 6 5 6 6 5 6 6

Self-Avoiding Walk Tree (Godsil 1981) G =( V , E ) T = T ��� ( G, v ) 1 v 1 4 2 3 4 3 2 6 6 4 3 5 σ Λ 5 6 6 5 6 6 5 6 6 5 for monomer-dimer: 6 6 P G [ v is matched | σ Λ ] = P T [ v is matched | σ Λ ]

Self-Avoiding Walk Tree (Weitz 2006) G =( V , E ) T = T ��� ( G, v ) 1 v 1 4 2 3 4 3 2 6 6 4 3 5 σ Λ 5 6 6 5 1 6 6 1 5 6 6 4 5 4 6 6 4 4 if cycle closing > cycle starting if cycle closing < cycle starting