Uniqueness, Spatial Mixing, and Approximation for Ferromagnetic 2-Spin Systems Heng Guo 1 and Pinyan Lu 2 1 Queen Mary, University of London 2 Shanghai University of Finance and Economics Paris, France Sep 08 2016 Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 1 / 19
Ising Model 0 1 Edge interaction 0 β 1 1 1 β β β 1 1 β 1 1 β β β 1 1 1 β β 1 Configuration σ : V → { 0, 1 } w ( σ ) = β mono ( σ ) π ( σ ) ∼ w ( σ ) Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 2 / 19
Ising Model 0 1 Edge interaction 0 β 1 1 1 β β β 1 1 β 1 1 β β β 1 1 1 β β 1 Configuration σ : V → { 0, 1 } w ( σ ) = β 8 π ( σ ) ∼ w ( σ ) Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 2 / 19
Ising Model 0 1 Edge interaction 0 β 1 1 1 β β β 1 1 β 1 1 β β β 1 1 1 β β 1 Configuration σ : V → { 0, 1 } w ( σ ) = β 0 = 1 π ( σ ) ∼ w ( σ ) Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 2 / 19
Ising Model 0 1 Edge interaction 0 β 1 1 1 β β β 1 1 β 1 1 β β β 1 1 1 β β 1 Partition function (normalizing factor): ∑ Z G ( β ) = w ( σ ) σ : V → { 0,1 } where w ( σ ) = β mono ( σ ) , mono ( σ ) is the number of monochromatic edges under σ . Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 2 / 19
2-State Spin System 0 1 Edge: Vertex: 0 β 1 0 1 1 1 β 1 1 More generally, three parameters β , γ , and λ . w ( σ ) = β m 0 ( σ ) γ m 1 ( σ ) λ n 0 ( σ ) m 0 ( σ ) : # of ( 0, 0 ) edges; m 1 ( σ ) : # of ( 1, 1 ) edges; n 0 ( σ ) : # of 0 vertices. ∑ Z G ( β , γ , λ ) = w ( σ ) σ : V → { 0,1 } Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 3 / 19
2-State Spin System [ 1 [ ] ] β 1 Edge: Vertex: 1 β 1 More generally, three parameters β , γ , and λ . w ( σ ) = β m 0 ( σ ) γ m 1 ( σ ) λ n 0 ( σ ) m 0 ( σ ) : # of ( 0, 0 ) edges; m 1 ( σ ) : # of ( 1, 1 ) edges; n 0 ( σ ) : # of 0 vertices. ∑ Z G ( β , γ , λ ) = w ( σ ) σ : V → { 0,1 } Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 3 / 19
2-State Spin System [ λ [ ] ] β 1 Edge: Vertex: 1 γ 1 More generally, three parameters β , γ , and λ . w ( σ ) = β m 0 ( σ ) γ m 1 ( σ ) λ n 0 ( σ ) m 0 ( σ ) : # of ( 0, 0 ) edges; m 1 ( σ ) : # of ( 1, 1 ) edges; n 0 ( σ ) : # of 0 vertices. ∑ Z G ( β , γ , λ ) = w ( σ ) σ : V → { 0,1 } Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 3 / 19
2-State Spin System [ λ [ ] ] β 1 Edge: Vertex: 1 γ 1 More generally, three parameters β , γ , and λ . w ( σ ) = β m 0 ( σ ) γ m 1 ( σ ) λ n 0 ( σ ) m 0 ( σ ) : # of ( 0, 0 ) edges; m 1 ( σ ) : # of ( 1, 1 ) edges; n 0 ( σ ) : # of 0 vertices. ∑ Z G ( β , γ , λ ) = w ( σ ) σ : V → { 0,1 } Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 3 / 19
2-State Spin System [ λ [ ] ] β 1 Edge: Vertex: 1 γ 1 More generally, three parameters β , γ , and λ . w ( σ ) = β m 0 ( σ ) γ m 1 ( σ ) λ n 0 ( σ ) m 0 ( σ ) : # of ( 0, 0 ) edges; m 1 ( σ ) : # of ( 1, 1 ) edges; n 0 ( σ ) : # of 0 vertices. ∑ Z G ( β , γ , λ ) = w ( σ ) σ : V → { 0,1 } Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 3 / 19
Examples [ 1 [ ] ] β 1 Ising model: and (no external field) 1 β 1 ∑ β mono ( σ ) Z G ( β ) = σ : V → { 0,1 } [ λ [ ] ] 0 1 Hardcore gas model: and (Weighted independent set) 1 1 1 ∑ λ | I | Z G ( β ) = Independent set I Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 4 / 19
Examples [ 1 [ ] ] β 1 Ising model: and (no external field) 1 β 1 ∑ β mono ( σ ) Z G ( β ) = σ : V → { 0,1 } [ λ [ ] ] 0 1 Hardcore gas model: and (Weighted independent set) 1 1 1 ∑ λ | I | Z G ( β ) = Independent set I Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 4 / 19
Approximate Counting Exact evaluating Z is # P -hard unless βγ = 1 or β = γ = 0 or λ = 0. Approximate the partition function Z . ▶ Fully Polynomial-time Randomized Approximation Scheme (FPRAS) and FPTAS: polynomial time in n and 1 ε (multiplicative error ε ). Approximating Z is equivalent to approximate marginal probabilities p v due to self-reducibility [Jerrum, Valiant, Vazirani 86]. Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 5 / 19
Approximate Counting Exact evaluating Z is # P -hard unless βγ = 1 or β = γ = 0 or λ = 0. Approximate the partition function Z . ▶ Fully Polynomial-time Randomized Approximation Scheme (FPRAS) and FPTAS: polynomial time in n and 1 ε (multiplicative error ε ). Approximating Z is equivalent to approximate marginal probabilities p v due to self-reducibility [Jerrum, Valiant, Vazirani 86]. Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 5 / 19
Approximate Counting Exact evaluating Z is # P -hard unless βγ = 1 or β = γ = 0 or λ = 0. Approximate the partition function Z . ▶ Fully Polynomial-time Randomized Approximation Scheme (FPRAS) and FPTAS: polynomial time in n and 1 ε (multiplicative error ε ). Approximating Z is equivalent to approximate marginal probabilities p v due to self-reducibility [Jerrum, Valiant, Vazirani 86]. Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 5 / 19
Ferromagnetic and Anti-ferromagnetic Edge Interaction [ ] β 1 1 γ If βγ = 1, then the 2-spin system is trivial. Ferromagnetic Ising: βγ > 1. Neighbours tend to have the same spin. Anti-ferromagnetic Ising: βγ < 1. Neighbours tend to have different spins. Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 6 / 19
Ferromagnetic and Anti-ferromagnetic Edge Interaction [ ] β 1 1 γ If βγ = 1, then the 2-spin system is trivial. Ferromagnetic Ising: βγ > 1. Neighbours tend to have the same spin. Anti-ferromagnetic Ising: βγ < 1. Neighbours tend to have different spins. Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 6 / 19
Ferromagnetic and Anti-ferromagnetic Edge Interaction [ ] β 1 1 γ If βγ = 1, then the 2-spin system is trivial. Ferromagnetic Ising: β = γ > 1. βγ > 1. Neighbours tend to have the same spin. Anti-ferromagnetic Ising: β = γ < 1. βγ < 1. Neighbours tend to have different spins. Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 6 / 19
Ferromagnetic and Anti-ferromagnetic Edge Interaction [ ] β 1 1 γ If βγ = 1, then the 2-spin system is trivial. Ferromagnetic Ising: βγ > 1. Neighbours tend to have the same spin. Anti-ferromagnetic Ising: βγ < 1. Neighbours tend to have different spins. Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 6 / 19
Ferromagnetic 2-Spin Systems Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 7 / 19
Previous Work FPRAS exists for ferromagnetic Ising models with consistent fields: β = γ > 1 and λ v ⩾ 1 (or ⩽ 1) for all v ∈ V [Jerrum, Sinclair 93]. Extended to λ v ⩽ γ β (if β ⩽ γ and βγ > 1) [Goldberg, Jerrum, Paterson 03], [Liu, Lu, Zhang 14]. Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 8 / 19
Previous Work FPRAS exists for ferromagnetic Ising models with consistent fields: β = γ > 1 and λ v ⩾ 1 (or ⩽ 1) for all v ∈ V [Jerrum, Sinclair 93]. Extended to λ v ⩽ γ β (if β ⩽ γ and βγ > 1) [Goldberg, Jerrum, Paterson 03], [Liu, Lu, Zhang 14]. Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 8 / 19
Main Theorem Theorem ) ∆ c / 2 ( γ If β ⩽ 1 ⩽ γ , βγ > 1 , and λ v ⩽ λ c = β 2 √ βγ where ∆ c = √ βγ − 1 , then FPTAS exists. ) ( ⌊ ∆ c ⌋ + 1 ) / 2 ( γ If we allow λ v > λ int c = , β then Z is #BIS-hard to approximate [Liu, Lu, Zhang 14]. #BIS is the complexity upper bound for all ferro 2-spin systems. Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 9 / 19
Ferro 2-Spin [ λ v [ ] ] β 1 Ferro 2-spin systems: Edge: Vertex: 1 γ 1 Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 10 / 19
Ferro 2-Spin [ λ v [ ] ] β 1 Ferro 2-spin systems: Edge: Vertex: 1 γ 1 For general graph G , assuming β ⩽ γ : FPRAS [LLZ14] γ λ β Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 10 / 19
Ferro 2-Spin [ λ v [ ] ] β 1 Ferro 2-spin systems: Edge: Vertex: 1 γ 1 For general graph G , assuming β ⩽ γ : FPRAS [LLZ14] #BIS-hard [LLZ14] γ λ int λ c β ) ( ⌊ ∆ c ⌋ + 1 ) / 2 2 √ βγ ( γ λ int c = , where ∆ c = √ βγ − 1 . β Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 10 / 19
Ferro 2-Spin [ λ v [ ] ] β 1 Ferro 2-spin systems: Edge: Vertex: 1 γ 1 For general graph G , assuming β ⩽ γ : FPRAS [LLZ14] #BIS-hard [LLZ14] γ λ c λ int CSM λ c β [G. Lu 16] ) ( ⌊ ∆ c ⌋ + 1 ) / 2 2 √ βγ ) ∆ c / 2 ( ( γ γ λ int c = , where ∆ c = √ βγ − 1 . λ c = β β Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 10 / 19
Ferro 2-Spin [ λ v [ ] ] β 1 Ferro 2-spin systems: Edge: Vertex: 1 γ 1 For general graph G , assuming β ⩽ γ : FPRAS [LLZ14] #BIS-hard [LLZ14] γ λ c λ int CSM λ c β [G. Lu 16] FPTAS (assume β ⩽ 1 ⩽ γ ) ) ( ⌊ ∆ c ⌋ + 1 ) / 2 2 √ βγ ) ∆ c / 2 ( ( γ γ λ int c = , where ∆ c = √ βγ − 1 . λ c = β β Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 10 / 19
Weitz’s Correlation Decay Algorithm Goal: calculate marginal probabilities using tree recursions. Replace a vertex of degree d with d copies. v R v = Pr ( v = 0 ) Pr ( v = 1 ) Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 11 / 19
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