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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

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SLIDE 1

Uniqueness, Spatial Mixing, and Approximation for Ferromagnetic 2-Spin Systems

Heng Guo1 and Pinyan Lu2

1Queen Mary, University of London 2Shanghai University of Finance and Economics

Paris, France Sep 08 2016

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 1 / 19

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SLIDE 2

Ising Model Edge interaction

0 1 0 β 1 1 1 β 1 β β 1 1 β β 1 1 β β 1 β 1 1 β

Configuration σ : V → {0, 1} w(σ) = βmono(σ) π(σ) ∼ w(σ)

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 2 / 19

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SLIDE 3

Ising Model Edge interaction

0 1 0 β 1 1 1 β 1 β β 1 1 β β 1 1 β β 1 β 1 1 β

Configuration σ : V → {0, 1} w(σ) = β8 π(σ) ∼ w(σ)

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 2 / 19

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SLIDE 4

Ising Model Edge interaction

0 1 0 β 1 1 1 β 1 β β 1 1 β β 1 1 β β 1 β 1 1 β

Configuration σ : V → {0, 1} w(σ) = β0 = 1 π(σ) ∼ w(σ)

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 2 / 19

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SLIDE 5

Ising Model Edge interaction

0 1 0 β 1 1 1 β 1 β β 1 1 β β 1 1 β β 1 β 1 1 β

Partition function (normalizing factor): ZG(β) = ∑

σ:V→{0,1}

w(σ) where w(σ) = βmono(σ), mono(σ) is the number of monochromatic edges under σ.

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 2 / 19

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SLIDE 6

2-State Spin System Edge:

0 1 0 β 1 1 1 β

Vertex:

0 1 1 1

More generally, three parameters β, γ, and λ.

w(σ) = βm0(σ)γm1(σ)λn0(σ) m0(σ): # of (0, 0) edges; m1(σ): # of (1, 1) edges; n0(σ): # of 0 vertices. ZG(β, γ, λ) = ∑

σ:V→{0,1}

w(σ)

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 3 / 19

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SLIDE 7

2-State Spin System Edge:

[

β 1 1 β

]

Vertex:

[ 1

1

]

More generally, three parameters β, γ, and λ.

w(σ) = βm0(σ)γm1(σ)λn0(σ) m0(σ): # of (0, 0) edges; m1(σ): # of (1, 1) edges; n0(σ): # of 0 vertices. ZG(β, γ, λ) = ∑

σ:V→{0,1}

w(σ)

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 3 / 19

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SLIDE 8

2-State Spin System Edge:

[

β 1 1 γ

]

Vertex:

[ λ

1

]

More generally, three parameters β, γ, and λ.

w(σ) = βm0(σ)γm1(σ)λn0(σ) m0(σ): # of (0, 0) edges; m1(σ): # of (1, 1) edges; n0(σ): # of 0 vertices. ZG(β, γ, λ) = ∑

σ:V→{0,1}

w(σ)

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 3 / 19

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SLIDE 9

2-State Spin System Edge:

[

β 1 1 γ

]

Vertex:

[ λ

1

]

More generally, three parameters β, γ, and λ.

w(σ) = βm0(σ)γm1(σ)λn0(σ) m0(σ): # of (0, 0) edges; m1(σ): # of (1, 1) edges; n0(σ): # of 0 vertices. ZG(β, γ, λ) = ∑

σ:V→{0,1}

w(σ)

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 3 / 19

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SLIDE 10

2-State Spin System Edge:

[

β 1 1 γ

]

Vertex:

[ λ

1

]

More generally, three parameters β, γ, and λ.

w(σ) = βm0(σ)γm1(σ)λn0(σ) m0(σ): # of (0, 0) edges; m1(σ): # of (1, 1) edges; n0(σ): # of 0 vertices. ZG(β, γ, λ) = ∑

σ:V→{0,1}

w(σ)

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 3 / 19

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SLIDE 11

Examples Ising model:

[

β 1 1 β

]

and

[ 1

1

]

(no external field) ZG(β) =

∑

σ:V→{0,1}

βmono(σ)

Hardcore gas model:

[

0 1 1 1

]

and

[ λ

1

]

(Weighted independent set) ZG(β) =

∑

Independent set I

λ|I|

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 4 / 19

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SLIDE 12

Examples Ising model:

[

β 1 1 β

]

and

[ 1

1

]

(no external field) ZG(β) =

∑

σ:V→{0,1}

βmono(σ)

Hardcore gas model:

[

0 1 1 1

]

and

[ λ

1

]

(Weighted independent set) ZG(β) =

∑

Independent set I

λ|I|

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 4 / 19

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SLIDE 13

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 pv due to self-reducibility [Jerrum, Valiant, Vazirani 86].

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 5 / 19

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SLIDE 14

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 pv due to self-reducibility [Jerrum, Valiant, Vazirani 86].

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 5 / 19

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SLIDE 15

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 pv due to self-reducibility [Jerrum, Valiant, Vazirani 86].

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 5 / 19

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SLIDE 16

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

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SLIDE 17

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

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SLIDE 18

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

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SLIDE 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

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SLIDE 20

Ferromagnetic 2-Spin Systems

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 7 / 19

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SLIDE 21

Previous Work

FPRAS exists for ferromagnetic Ising models with consistent fields: β = γ > 1 and λv ⩾ 1 (or ⩽ 1) for all v ∈ V [Jerrum, Sinclair 93]. Extended to λv ⩽ 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

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SLIDE 22

Previous Work

FPRAS exists for ferromagnetic Ising models with consistent fields: β = γ > 1 and λv ⩾ 1 (or ⩽ 1) for all v ∈ V [Jerrum, Sinclair 93]. Extended to λv ⩽ 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

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SLIDE 23

Main Theorem Theorem If β ⩽ 1 ⩽ γ, βγ > 1, and λv ⩽ λc =

(

γ β

)∆c/2

where ∆c =

2√βγ √βγ−1, then FPTAS exists.

If we allow λv > λint

c =

(

γ β

)(⌊∆c⌋+1)/2

, 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

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SLIDE 24

Ferro 2-Spin

Ferro 2-spin systems: Edge: [

β 1 1 γ

] Vertex: [ λv

1

]

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 10 / 19

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SLIDE 25

Ferro 2-Spin

Ferro 2-spin systems: Edge: [

β 1 1 γ

] Vertex: [ λv

1

] For general graph G, assuming β ⩽ γ: λ

γ β

FPRAS [LLZ14]

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 10 / 19

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SLIDE 26

Ferro 2-Spin

Ferro 2-spin systems: Edge: [

β 1 1 γ

] Vertex: [ λv

1

] For general graph G, assuming β ⩽ γ: λ

γ β

λint

c

#BIS-hard [LLZ14] FPRAS [LLZ14] λint

c =

(

γ β

)(⌊∆c⌋+1)/2 , where ∆c =

2√βγ √βγ−1. Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 10 / 19

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SLIDE 27

Ferro 2-Spin

Ferro 2-spin systems: Edge: [

β 1 1 γ

] Vertex: [ λv

1

] For general graph G, assuming β ⩽ γ: λ

γ β

λc λint

c

#BIS-hard [LLZ14] FPRAS [LLZ14] CSM [G. Lu 16] λint

c =

(

γ β

)(⌊∆c⌋+1)/2 , where ∆c =

2√βγ √βγ−1.

λc = (

γ β

)∆c/2

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 10 / 19

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SLIDE 28

Ferro 2-Spin

Ferro 2-spin systems: Edge: [

β 1 1 γ

] Vertex: [ λv

1

] For general graph G, assuming β ⩽ γ: λ

γ β

λc λint

c

#BIS-hard [LLZ14] FPRAS [LLZ14] CSM [G. Lu 16] FPTAS

(assume β ⩽ 1 ⩽ γ) λint

c =

(

γ β

)(⌊∆c⌋+1)/2 , where ∆c =

2√βγ √βγ−1.

λc = (

γ β

)∆c/2

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 10 / 19

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SLIDE 29

Weitz’s Correlation Decay Algorithm Goal: calculate marginal probabilities using tree recursions.

Replace a vertex of degree d with d copies. Rv = Pr(v = 0) Pr(v = 1)

v

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 11 / 19

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SLIDE 30

Weitz’s Correlation Decay Algorithm Goal: calculate marginal probabilities using tree recursions.

Replace a vertex of degree d with d copies. Rv = Pr(v = 0) Pr(v = 1) = Pr(v1 = 0, . . . , vd = 0) Pr(v1 = 1, . . . , vd = 1)

v

⇓

v1 v2 v3 v4

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 11 / 19

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SLIDE 31

Weitz’s Correlation Decay Algorithm Goal: calculate marginal probabilities using tree recursions.

Replace a vertex of degree d with d copies. Rv = Pr(v = 0) Pr(v = 1) = Pr(v1 = 0, . . . , vd = 0) Pr(v1 = 1, . . . , vd = 1) = Pr(0000) Pr(0001) · Pr(0001) Pr(0011) · Pr(0011) Pr(0111) · Pr(0111) Pr(1111)

v

⇓

v1 v2 v3 v4

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 11 / 19

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SLIDE 32

Weitz’s Correlation Decay Algorithm Goal: calculate marginal probabilities using tree recursions.

Replace a vertex of degree d with d copies. Rv = Pr(v = 0) Pr(v = 1) = Pr(v1 = 0, . . . , vd = 0) Pr(v1 = 1, . . . , vd = 1) = Pr(0000) Pr(0001) · Pr(0001) Pr(0011) · Pr(0011) Pr(0111) · Pr(0111) Pr(1111) Each term Pr(0011)

Pr(0111) can be viewed as the marginal ratio of

vi conditioned on a certain configuration of other vj’s.

v

⇓

v1 v2 v3 v4

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 11 / 19

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SLIDE 33

Self-Avoiding Walk (SAW) Tree

SAW tree is essentially the tree of self-avoiding walks originating at v except that the vertices closing a cycle are also included in the tree.

▶ Cycle-closing vertices are fixed according to the rule in the last slide.

Do the tree recursion to calculate pv.

a b c d e f

⇒

a b c d e f d c f e a c c a e f f e c c

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 12 / 19

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SLIDE 34

Correlation Decay SAW tree has exponential size in general.

▶ Truncate the recursion within logarithmic depth. ▶ How much error does the truncation incur?

a b c d e f d c f e a c c a e f f e c c

⇒

a b c d e f d c f e a c c a e f f e c c

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 13 / 19

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SLIDE 35

Spatial Mixing in Trees Weak Spatial Mixing:

a b c d e f d c f e a c c a e f f e c c

v.s.

a b c d e f d c f e a c c a e f f e c c

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 14 / 19

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SLIDE 36

Spatial Mixing in Trees Strong Spatial Mixing:

a b c d e f d c f e a c c a e f f e c c

v.s.

a b c d e f d c f e a c c a e f f e c c

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 14 / 19

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SLIDE 37

Conditional Spatial Mixing

If λv < λc for all v, conditional spatial mixing holds in arbitrary trees: Instead of worst case configurations in SSM, we only allow partial configurations that are dominated by the product measure of isolated vertices (pv ⩽

λ 1+λ).

(All vertices are leaning towards the good spin.)

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 15 / 19

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SLIDE 38

Conditional Spatial Mixing

If λv < λc for all v, conditional spatial mixing holds in arbitrary trees: Instead of worst case configurations in SSM, we only allow partial configurations that are dominated by the product measure of isolated vertices (pv ⩽

λ 1+λ).

(All vertices are leaning towards the good spin.) SSM:

a b c d e f d c f e a c c a e f f e c c

v.s.

a b c d e f d c f e a c c a e f f e c c

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 15 / 19

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SLIDE 39

Conditional Spatial Mixing

If λv < λc for all v, conditional spatial mixing holds in arbitrary trees: Instead of worst case configurations in SSM, we only allow partial configurations that are dominated by the product measure of isolated vertices (pv ⩽

λ 1+λ).

(All vertices are leaning towards the good spin.) Conditional spatial mixing:

a b c d e f d c f e a c c a e f f e c c

v.s.

a b c d e f d c f e a c c a e f f e c c

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 15 / 19

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SLIDE 40

Pruning If β ⩽ 1 < γ, in the SAW tree, we may first remove “bad” pinnings, the effective field is smaller (better).

a b c d e f d c f e a c c a e f f e c c

v.s.

a b c d e f d c f e a c c a e f f e c c

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 16 / 19

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SLIDE 41

Pruning If β ⩽ 1 < γ, in the SAW tree, we may first remove “bad” pinnings, the effective field is smaller (better).

a b c d e f d c f e a c c a e f f e c c

v.s.

a b c d e f d c f e a c c a e f f e c c

CSM ⇒ SSM

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 16 / 19

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SLIDE 42

What about β > 1? If β > 1, then pruning fails. In fact, there is no λ such that SSM holds for general trees.

a b c d e f d c f e a c c a e f f e c c

⇒

a b c d e f d c f e a c c a e f f e c c

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 17 / 19

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SLIDE 43

What about β > 1? If β > 1, then pruning fails. In fact, there is no λ such that SSM holds for general trees.

a b c d e f d c f e a c c a e f f e c c

⇒

a b c d e f d c f e a c c a e f f e c c

However, if λv ⩽ λc, then pv ⩽

λ 1+λ for any graph G.

FPTAS without SSM?

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 17 / 19

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SLIDE 44

The Exact Threshold? Our result is tight up to an integrality gap. However, neither λc nor λint

c is the right bound.

There exists a small interval beyond λc where FPTAS still exists.

▶ Degrees have to be integers.

There is a λ < λint

c such that SSM fails (in an irregular tree).

WSM (in T∆) ̸⇒ SSM

(even if β ⩽ 1 < γ) This is in contrast to the anti-ferro case.

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 18 / 19

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SLIDE 45

The Exact Threshold? Our result is tight up to an integrality gap. However, neither λc nor λint

c is the right bound.

There exists a small interval beyond λc where FPTAS still exists.

▶ Degrees have to be integers.

There is a λ < λint

c such that SSM fails (in an irregular tree).

WSM (in T∆) ̸⇒ SSM

(even if β ⩽ 1 < γ) This is in contrast to the anti-ferro case.

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 18 / 19

slide-46
SLIDE 46

The Exact Threshold? Our result is tight up to an integrality gap. However, neither λc nor λint

c is the right bound.

There exists a small interval beyond λc where FPTAS still exists.

▶ Degrees have to be integers.

There is a λ < λint

c such that SSM fails (in an irregular tree).

WSM (in T∆) ̸⇒ SSM

(even if β ⩽ 1 < γ) This is in contrast to the anti-ferro case.

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 18 / 19

slide-47
SLIDE 47

Open Questions Ferro 2-spin systems: FPTAS for 1 < β ⩽ γ, λv < λc?

▶ Conditional spatial mixing for graphs instead of trees.

(This implies FPTAS for, say, planar graphs.)

Avoid the gadget gap in the hardness proof.

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 19 / 19

slide-48
SLIDE 48

Open Questions Ferro 2-spin systems: FPTAS for 1 < β ⩽ γ, λv < λc?

▶ Conditional spatial mixing for graphs instead of trees.

(This implies FPTAS for, say, planar graphs.)

Avoid the gadget gap in the hardness proof.

Thank You!

Heng Guo (QMUL) Ferro 2-Spin Systems RANDOM 2016 19 / 19