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

Spatial Mixing and the Connective Constant:

• ptimal bounds

Yitong Yin Nanjing University Joint work with: Alistair Sinclair (UC Berkeley) Piyush Srivastava (UC Berkeley) Daniel Štefankovič (Rochester)

G = (V, E) undirected graph

approximately counting # of

• f G

matchings independent sets

{ }

almost uniformly sampling a

• f G

matching independent set

{ }

G = (V, E) undirected graph

approximately counting # of

• f G

matchings independent sets

{ }

almost uniformly sampling a

• f G

matching independent set

{ }

computationally equivalent

Zλ(G) = X

M∈M(G)

λ|M|

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

Zλ(G) = X

M∈M(G)

λ|M|

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

µ(M) = λ|M| Zλ(G)

Gibbs distribution

Zλ(G) = X

I∈I(G)

λ|I|

Zλ(G) = X

M∈M(G)

λ|M|

M(G) I(G) monomer-dimer model: hardcore model: G = (V, E) undirected graph set of all matchings partition function partition function set of all independent sets

µ(I) = λ|I| Zλ(G)

µ(M) = λ|M| Zλ(G)

Gibbs distribution Gibbs distribution

Known Results

computing the partition function

Known Results

• monomer-dimer (matching)

computing the partition function

Known Results

• monomer-dimer (matching)
• FPRAS by MCMC [Jerrum, Sinclair 1989]

computing the partition function

Known Results

• monomer-dimer (matching)
• FPRAS by MCMC [Jerrum, Sinclair 1989]
• FPTAS for graphs with bounded max-degree [Bayati et

al, STOC 2007]

computing the partition function

Known Results

• monomer-dimer (matching)
• FPRAS by MCMC [Jerrum, Sinclair 1989]
• FPTAS for graphs with bounded max-degree [Bayati et

al, STOC 2007]

• hardcore (independent set)

computing the partition function

Known Results

• 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]

computing the partition function

Known Results

• 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<λc(d-1)

[Weitz, STOC 2006]

computing the partition function

Known Results

• 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<λc(d-1)

[Weitz, STOC 2006]

uniqueness threshold:

computing the partition function

λc(d) = dd (d − 1)d+1

Known Results

• 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<λc(d-1)

[Weitz, STOC 2006]

• inapproxamable if λ>λc(d-1) [Sly, FOCS 2010; Sly, Sun,

FOCS 2012]

uniqueness threshold:

computing the partition function

λc(d) = dd (d − 1)d+1

Known Results

• 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<λc(d-1)

[Weitz, STOC 2006]

• inapproxamable if λ>λc(d-1) [Sly, FOCS 2010; Sly, Sun,

FOCS 2012]

uniqueness threshold:

spatial mixing

computing the partition function

λc(d) = dd (d − 1)d+1

Connective Constants

N(v, `) : number of paths of length l starting from v

[Madras, Slade 1996]

Connective Constants

∆(G) = sup

v∈V

lim sup

`→∞

N(v, `)1/`

N(v, `) : number of paths of length l starting from v for an infinite graph G: connective constant

[Madras, Slade 1996]

Connective Constants

∆(G) = sup

v∈V

lim sup

`→∞

N(v, `)1/`

N(v, `) : number of paths of length l starting from v for an infinite graph G: connective constant can be similarly defined for a family of finite graphs G

[Madras, Slade 1996]

Connective Constants

∆(G) = sup

v∈V

lim sup

`→∞

N(v, `)1/`

N(v, `) : number of paths of length l starting from v for an infinite graph G: connective constant can be similarly defined for a family of finite graphs G

[Madras, Slade 1996]

the connective constant represents the average growth rate of number of paths from a vertex

Connective Constants

∆(G) = sup

v∈V

lim sup

`→∞

N(v, `)1/`

N(v, `) : number of paths of length l starting from v for an infinite graph G: connective constant can be similarly defined for a family of finite graphs G

[Madras, Slade 1996]

the connective constant represents the average growth rate of number of paths from a vertex for honeycomb lattice

q 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<λc(d-1)

[Weitz, STOC 2006]

• FPTAS for graphs with connective constant Δ when

λ<λc(Δ).

λc(∆) = ∆∆ (∆ − 1)∆+1

uniqueness threshold:

Zλ(G) = X

I∈I(G)

λ|I|

Zλ(G) = X

M∈M(G)

λ|M|

hardcore partition functions:

µ(I) = λ|I| Zλ(G)

µ(M) = λ|M| Zλ(G)

Gibbs distributions: monomer-dimer marginal probabilities:

Pr[v is matched | σΛ]

Pr[v is occupied | σΛ]

σΛ : configuration of being matched/unmatched or

• ccupied/unoccupied for vertices in Λ⊂V

Zλ(G) = X

I∈I(G)

λ|I|

Zλ(G) = X

M∈M(G)

λ|M|

hardcore partition functions:

µ(I) = λ|I| Zλ(G)

µ(M) = λ|M| Zλ(G)

Gibbs distributions: monomer-dimer marginal probabilities:

Pr[v is matched | σΛ]

Pr[v is occupied | σΛ]

σΛ : configuration of being matched/unmatched or

• ccupied/unoccupied for vertices in Λ⊂V

by self-reduction:

(Jerrum-Valiant-Vazirani)

efficient approximation of marginal probabilities implies efficient approximation of partition function

Spatial Mixing (Decay of Correlation)

R G v t weak spatial mixing (WSM):

error < exp (-t) Pr[c(v) = x | σ∂R] ≈ Pr[c(v) = x | τ∂R] ∂R

Spatial Mixing (Decay of Correlation)

R G v t weak spatial mixing (WSM):

error < exp (-t) Pr[c(v) = x | σ∂R] ≈ Pr[c(v) = x | τ∂R] ∂R

uniqueness threshold: WSM in d-regular tree

Spatial Mixing (Decay of Correlation)

R G v t weak spatial mixing (WSM):

error < exp (-t) Pr[c(v) = x | σ∂R] ≈ Pr[c(v) = x | τ∂R] ∂R

Spatial Mixing (Decay of Correlation)

R G v t

Λ

weak spatial mixing (WSM): strong spatial mixing (SSM):

error < exp (-t) Pr[c(v) = x | σ∂R] ≈ Pr[c(v) = x | τ∂R] Pr[c(v) = x | σ∂R, σΛ] ≈ Pr[c(v) = x | τ∂R, σΛ] ∂R

Spatial Mixing (Decay of Correlation)

R G v t

Λ

weak spatial mixing (WSM): strong spatial mixing (SSM):

error < exp (-t) Pr[c(v) = x | σΛ] is approximable by local information

SSM: the value of

Pr[c(v) = x | σ∂R] ≈ Pr[c(v) = x | τ∂R] Pr[c(v) = x | σ∂R, σΛ] ≈ Pr[c(v) = x | τ∂R, σΛ] ∂R

Self-Avoiding Walk Tree

(Godsil 1981)

1 2 3 4 5 6

G=(V,E) v

Self-Avoiding Walk Tree

(Godsil 1981)

1 2 3 4 5 6 1

G=(V,E) v

Self-Avoiding Walk Tree

(Godsil 1981)

1 2 3 4 5 6 2 6 5 5 6 4 3 3 5 6 5 6 4 1

G=(V,E) v

T = T(G, v)

Self-Avoiding Walk Tree

(Godsil 1981)

1 2 3 4 5 6 2 6 5 5 6 4 3 3 5 6 5 6 4 1

G=(V,E) v

T = T(G, v) 6 σΛ

Self-Avoiding Walk Tree

(Godsil 1981)

1 2 3 4 5 6 2 6 5 5 6 4 3 3 5 6 5 6 4 1

G=(V,E) v

T = T(G, v) 6 6 6 6 6 σΛ

Self-Avoiding Walk Tree

(Godsil 1981)

1 2 3 4 5 6 2 6 5 5 6 4 3 3 5 6 5 6 4 1

G=(V,E) v

T = T(G, v) 6 6 6 6 6 σΛ

PG[v is matched | σΛ] =PT [v is matched | σΛ]

for monomer-dimer:

1

Self-Avoiding Walk Tree

(Weitz 2006)

1 2 3 4 5 6 2 4 4 1 4 4 6 5 5 6 4 3 3 5 6 5 6 4 1

G=(V,E) v

if cycle closing > cycle starting if cycle closing < cycle starting

T = T(G, v) 6 6 6 6 6 σΛ

1

Self-Avoiding Walk Tree

(Weitz 2006)

1 2 3 4 5 6 2 4 4 1 4 4 6 5 5 6 4 3 3 5 6 5 6 4 1

G=(V,E) v

if cycle closing > cycle starting if cycle closing < cycle starting

T = T(G, v) 6 6 6 6 6 σΛ for hardcore: PG[v is occupied | σΛ] =PT [v is occupied | σΛ]

Tree Recursions

6 6 6 6 T = T(G, v) x

xi x1

xd

Tree Recursions

6 6 6 6 T = T(G, v) x

xi x1

xd

monomer-dimer model:

x = f(~ x) = 1 1 + Pd

i=1 xi

x: marginal probability of being unmatched

Tree Recursions

6 6 6 6 T = T(G, v) x

xi x1

xd

monomer-dimer model: hardcore model:

x = f(~ x) = 1 1 + Pd

i=1 xi

x = f(~ x) =

d

Y

i=1

1 1 + xi

x: marginal probability of being unmatched x: ratio between marginal probabilities

• f being occupied and unoccupied

Tree Recursions

6 6 6 6 T = T(G, v) x

xi x1

xd

monomer-dimer model: hardcore model:

x = f(~ x) = 1 1 + Pd

i=1 xi

x = f(~ x) =

d

Y

i=1

1 1 + xi

x: marginal probability of being unmatched x: ratio between marginal probabilities

• f being occupied and unoccupied

truncated

Tree Recursions

6 6 6 6 T = T(G, v) x

xi x1

xd

monomer-dimer model: hardcore model:

x = f(~ x) = 1 1 + Pd

i=1 xi

x = f(~ x) =

d

Y

i=1

1 1 + xi

x: marginal probability of being unmatched x: ratio between marginal probabilities

• f being occupied and unoccupied

truncated

initial errors: ~

✏ error ✏

Tree Recursions

6 6 6 6 T = T(G, v) x

xi x1

xd

monomer-dimer model: hardcore model:

x = f(~ x) = 1 1 + Pd

i=1 xi

x = f(~ x) =

d

Y

i=1

1 1 + xi

x: marginal probability of being unmatched x: ratio between marginal probabilities

• f being occupied and unoccupied

` connective constant:

∆` =

d

X

i=1

∆(`−1)

i

truncated

initial errors: ~

✏ error ✏

# of l-level nodes:

Spatial Mixing

6 6 6 6 T = T(G, v) x

xi x1

xd

` connective constant:

∆` =

d

X

i=1

∆(`−1)

i

truncated

initial errors: ~

✏ error ✏

✏i ✏1 ✏d

Spatial Mixing

6 6 6 6 T = T(G, v) x

xi x1

xd

` connective constant:

∆` =

d

X

i=1

∆(`−1)

i

truncated

initial errors: ~

✏ error ✏ Mean Value Thm:

✏ ≤

d

X

i=1

• @f(~

x) @xi

• ✏i

✏i ✏1 ✏d

SSM: ✏ = exp(−Ω(`))

Spatial Mixing

6 6 6 6 T = T(G, v) x

xi x1

xd

` connective constant:

∆` =

d

X

i=1

∆(`−1)

i

truncated

initial errors: ~

✏ error ✏ Mean Value Thm:

✏ ≤

d

X

i=1

• @f(~

x) @xi

• ✏i

✏i ✏1 ✏d

error ε t

ideal real

SSM: ✏ = exp(−Ω(`))

The Potential Method

error error ✏ error error δ

φ

• riginal:

potential:

x = f(~ x) xi ✏i yi = φ(xi) y = φ(x) y = g(~ y) δi yi

The Potential Method

error error ✏ error error δ

φ

• riginal:

potential:

g

new recursion

x = f(~ x) xi ✏i yi = φ(xi) y = φ(x) y = g(~ y) δi y = g(~ y) = (f(−1(y1), −1(y2), . . . , −1(yd)))) yi

The Potential Method

error error ✏ error error δ

φ

• riginal:

potential:

g

new recursion

by Mean Value Thm:

Φ(x) = d φ(x) d x

let

x = f(~ x) xi ✏i yi = φ(xi) y = φ(x) y = g(~ y) δi y = g(~ y) = (f(−1(y1), −1(y2), . . . , −1(yd))))

≤

d

X

i=1

• @f(~

x) @xi

• Φ(f(~

x)) Φ(xi) i

yi

The Potential Method

error error ✏ error error δ

φ

• riginal:

potential:

g

new recursion

with good choice of potential function φ :

error ε t error δ t

φ

• riginal

world potential world by Mean Value Thm:

Φ(x) = d φ(x) d x

let

x = f(~ x) xi ✏i yi = φ(xi) y = φ(x) y = g(~ y) δi y = g(~ y) = (f(−1(y1), −1(y2), . . . , −1(yd))))

≤

d

X

i=1

• @f(~

x) @xi

• Φ(f(~

x)) Φ(xi) i

yi

f(~ x) = 1 1 + Pd

i=1 xi

δ

δi

∆i

`

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

recursion:

∆` =

d

X

i=1

∆(`−1)

i

d

initial δ’s

for monomer-dimer model (matchings):

f(~ x) = 1 1 + Pd

i=1 xi

δ = Φ(f)

d

X

i=1

• ∂f

∂xi

• δi

Φ(xi)

δ

δi

∆i

`

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

recursion:

∆` =

d

X

i=1

∆(`−1)

i

d

initial δ’s

for monomer-dimer model (matchings):

f(~ x) = 1 1 + Pd

i=1 xi

Φ(x) = 1 x

δ = Φ(f)

d

X

i=1

• ∂f

∂xi

• δi

Φ(xi) = λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

δ

δi

choose

∆i

`

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

recursion:

∆` =

d

X

i=1

∆(`−1)

i

d

initial δ’s

for monomer-dimer model (matchings):

f(~ x) = 1 1 + Pd

i=1 xi

Φ(x) = 1 x

δ = Φ(f)

d

X

i=1

• ∂f

∂xi

• δi

Φ(xi) = λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

δ

δi

choose

∆i

`

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

recursion:

∆` =

d

X

i=1

∆(`−1)

i

d

Goal:

initial δ = O(1) arbitrary

initial δ’s

as long as Δ = O(1) = exp(−Ω(`)) final ~ x ∈ [0, 1]d for monomer-dimer model (matchings):

= λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

δ

δi

∆i

`

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

d

initial δ’s

δ

= λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

δ

δi

∆i

`

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

δ effective degree:

= λ ˆ d 1 + λ ˆ d

d

X

i=1

xi ˆ d δi

ˆ d

ˆ d ,

d

X

i=1

xi

= λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

δ

δi

∆i

`

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

δ effective degree:

convex combination

• f δi’s

= λ ˆ d 1 + λ ˆ d

d

X

i=1

xi ˆ d δi

ˆ d

ˆ d ,

d

X

i=1

xi

= λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

δ

δi

∆i

`

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

δ effective degree:

convex combination

• f δi’s

= λ ˆ d 1 + λ ˆ d

d

X

i=1

αiδi

ˆ d αi , xi ˆ d = xi Pd

i=1 xi

ˆ d ,

d

X

i=1

xi

= λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

δ

δi

∆i

`

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

δ effective degree:

δi ≤ ✓ λ∆i 1 + λ∆i ◆`−1

I.H.:

convex combination

• f δi’s

= λ ˆ d 1 + λ ˆ d

d

X

i=1

αiδi

ˆ d αi , xi ˆ d = xi Pd

i=1 xi

ˆ d ,

d

X

i=1

xi

= λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

δ

δi

∆i

`

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

δ effective degree:

δi ≤ ✓ λ∆i 1 + λ∆i ◆`−1

I.H.:

convex combination

• f δi’s

= λ ˆ d 1 + λ ˆ d

d

X

i=1

αiδi

ˆ d

≤ λ ˆ d 1 + λ ˆ d !

d

X

i=1

αi ✓ λ∆i 1 + λ∆i ◆`−1

αi , xi ˆ d = xi Pd

i=1 xi

ˆ d ,

d

X

i=1

xi

= λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

δ

δi

∆i

`

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

δ effective degree:

δi ≤ ✓ λ∆i 1 + λ∆i ◆`−1

I.H.:

convex combination

• f δi’s

= λ ˆ d 1 + λ ˆ d

d

X

i=1

αiδi

ˆ d

≤ λ ˆ d 1 + λ ˆ d !

d

X

i=1

αi ✓ λ∆i 1 + λ∆i ◆`−1

αi , xi ˆ d = xi Pd

i=1 xi

ˆ ∆(`−1) ,

d

X

i=1

αi∆(`−1)

i

effective connective constant:

ˆ d ,

d

X

i=1

xi

= λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

δ

δi

∆i

`

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

δ effective degree:

δi ≤ ✓ λ∆i 1 + λ∆i ◆`−1

I.H.:

convex combination

• f δi’s

= λ ˆ d 1 + λ ˆ d

d

X

i=1

αiδi

ˆ d

≤ λ ˆ d 1 + λ ˆ d !

d

X

i=1

αi ✓ λ∆i 1 + λ∆i ◆`−1

αi , xi ˆ d = xi Pd

i=1 xi

ˆ ∆(`−1) ,

d

X

i=1

αi∆(`−1)

i

effective connective constant:

≤ λ ˆ d 1 + λ ˆ d ! λ ˆ ∆ 1 + λ ˆ ∆ !`−1

Jensen’s inequality

ˆ d ,

d

X

i=1

xi

= λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

δ `

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

δ effective degree:

δi ≤ ✓ λ∆i 1 + λ∆i ◆`−1

I.H.:

convex combination

• f δi’s

= λ ˆ d 1 + λ ˆ d

d

X

i=1

αiδi

ˆ d

≤ λ ˆ d 1 + λ ˆ d !

d

X

i=1

αi ✓ λ∆i 1 + λ∆i ◆`−1

αi , xi ˆ d = xi Pd

i=1 xi

ˆ ∆(`−1) ,

d

X

i=1

αi∆(`−1)

i

effective connective constant:

≤ λ ˆ d 1 + λ ˆ d ! λ ˆ ∆ 1 + λ ˆ ∆ !`−1

Jensen’s inequality

ˆ ∆ ˆ ∆ ˆ ∆

ˆ d ,

d

X

i=1

xi

δ `

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

δ effective degree:

δi ≤ ✓ λ∆i 1 + λ∆i ◆`−1

I.H.:

ˆ d αi , xi ˆ d = xi Pd

i=1 xi

ˆ ∆(`−1) ,

d

X

i=1

αi∆(`−1)

i

effective connective constant:

≤ λ ˆ d 1 + λ ˆ d ! λ ˆ ∆ 1 + λ ˆ ∆ !`−1 ˆ ∆ ˆ ∆ ˆ ∆

ˆ d ,

d

X

i=1

xi

δ `

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

δ effective degree:

δi ≤ ✓ λ∆i 1 + λ∆i ◆`−1

I.H.:

ˆ d αi , xi ˆ d = xi Pd

i=1 xi

ˆ ∆(`−1) ,

d

X

i=1

αi∆(`−1)

i

effective connective constant:

≤ λ ˆ d 1 + λ ˆ d ! λ ˆ ∆ 1 + λ ˆ ∆ !`−1 ˆ ∆ ˆ ∆ ˆ ∆

≤ B @ λ ⇣ ˆ d ˆ ∆`−1⌘1/` 1 + λ ⇣ ˆ d ˆ ∆`−1 ⌘1/` 1 C A

`

Jensen’s inequality

ˆ d ,

d

X

i=1

xi

δ `

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

δ effective degree:

δi ≤ ✓ λ∆i 1 + λ∆i ◆`−1

I.H.:

ˆ d αi , xi ˆ d = xi Pd

i=1 xi

ˆ ∆(`−1) ,

d

X

i=1

αi∆(`−1)

i

effective connective constant:

≤ λ ˆ d 1 + λ ˆ d ! λ ˆ ∆ 1 + λ ˆ ∆ !`−1 ˆ ∆ ˆ ∆ ˆ ∆

≤ B @ λ ⇣ ˆ d ˆ ∆`−1⌘1/` 1 + λ ⇣ ˆ d ˆ ∆`−1 ⌘1/` 1 C A

`

Jensen’s inequality

ˆ d ,

d

X

i=1

xi

geometric average

δ `

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

δ effective degree:

δi ≤ ✓ λ∆i 1 + λ∆i ◆`−1

I.H.:

ˆ d αi , xi ˆ d = xi Pd

i=1 xi

ˆ ∆(`−1) ,

d

X

i=1

αi∆(`−1)

i

effective connective constant:

≤ λ ˆ d 1 + λ ˆ d ! λ ˆ ∆ 1 + λ ˆ ∆ !`−1 ˆ ∆ ˆ ∆ ˆ ∆

≤ B @ λ ⇣ ˆ d ˆ ∆`−1⌘1/` 1 + λ ⇣ ˆ d ˆ ∆`−1 ⌘1/` 1 C A

`

Jensen’s inequality

ˆ d ,

d

X

i=1

xi

≤ ✓ λ∆ 1 + λ∆ ◆`

geometric average

Goal:

δ `

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

δ effective degree:

δi ≤ ✓ λ∆i 1 + λ∆i ◆`−1

I.H.:

ˆ d αi , xi ˆ d = xi Pd

i=1 xi

ˆ ∆(`−1) ,

d

X

i=1

αi∆(`−1)

i

effective connective constant:

≤ λ ˆ d 1 + λ ˆ d ! λ ˆ ∆ 1 + λ ˆ ∆ !`−1 ˆ ∆ ˆ ∆ ˆ ∆

≤ B @ λ ⇣ ˆ d ˆ ∆`−1⌘1/` 1 + λ ⇣ ˆ d ˆ ∆`−1 ⌘1/` 1 C A

`

Jensen’s inequality

ˆ d ˆ ∆`−1 ≤ ∆`

ˆ d ,

d

X

i=1

xi

≤ ✓ λ∆ 1 + λ∆ ◆`

geometric average

Goal:

• nly need to verify:

δ `

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

effective degree:

ˆ d αi , xi ˆ d = xi Pd

i=1 xi

ˆ ∆(`−1) ,

d

X

i=1

αi∆(`−1)

i

effective connective constant:

ˆ ∆ ˆ ∆ ˆ ∆

ˆ d ˆ ∆`−1 ≤ ∆`

ˆ d ,

d

X

i=1

xi

• nly need to verify:

δ `

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

effective degree:

ˆ d αi , xi ˆ d = xi Pd

i=1 xi

ˆ ∆(`−1) ,

d

X

i=1

αi∆(`−1)

i

effective connective constant:

ˆ ∆ ˆ ∆ ˆ ∆

ˆ d ˆ ∆`−1 ≤ ∆`

ˆ d ,

d

X

i=1

xi

• nly need to verify:

d X

i=1

xi ! d X

i=1

xi Pd

i=1 xi

∆`−1

i

! =

δ `

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

effective degree:

ˆ d αi , xi ˆ d = xi Pd

i=1 xi

ˆ ∆(`−1) ,

d

X

i=1

αi∆(`−1)

i

effective connective constant:

ˆ ∆ ˆ ∆ ˆ ∆

ˆ d ˆ ∆`−1 ≤ ∆`

ˆ d ,

d

X

i=1

xi

• nly need to verify:

d

X

i=1

xi∆(`−1)

i

=

δ `

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

effective degree:

ˆ d αi , xi ˆ d = xi Pd

i=1 xi

ˆ ∆(`−1) ,

d

X

i=1

αi∆(`−1)

i

effective connective constant:

ˆ ∆ ˆ ∆ ˆ ∆

ˆ d ˆ ∆`−1 ≤ ∆`

ˆ d ,

d

X

i=1

xi

• nly need to verify:

=

d

X

i=1

∆(`−1)

i

d

X

i=1

xi∆(`−1)

i

=

δ `

connective constants:

∆ ∆i

(for tree) (for i-th subtree)

∆` =

d

X

i=1

∆(`−1)

i

initial δ’s

effective degree:

ˆ d αi , xi ˆ d = xi Pd

i=1 xi

ˆ ∆(`−1) ,

d

X

i=1

αi∆(`−1)

i

effective connective constant:

ˆ ∆ ˆ ∆ ˆ ∆

ˆ d ˆ ∆`−1 ≤ ∆`

ˆ d ,

d

X

i=1

xi

• nly need to verify:

=

d

X

i=1

∆(`−1)

i

d

X

i=1

xi∆(`−1)

i

=

xi ∈ [0, 1]

recall:

SSM for matchings

6 6 6 6 T = T(G, v) x

xi x1

xd

` connective constant:

∆` =

d

X

i=1

∆(`−1)

i

truncated

initial errors:

error δ

~

• δ ≤

✓ λ∆ 1 + λ∆ ◆`

δ = λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

error:

SSM for matchings

6 6 6 6 T = T(G, v) x

xi x1

xd

` connective constant:

∆` =

d

X

i=1

∆(`−1)

i

truncated

initial errors:

error δ

~

• δ ≤

✓ λ∆ 1 + λ∆ ◆`

= exp(−Ω(`)) as long as Δ = O(1)

δ = λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

error:

SSM for matchings

6 6 6 6 T = T(G, v) x

xi x1

xd

` connective constant:

∆` =

d

X

i=1

∆(`−1)

i

truncated

initial errors:

error δ

~

• δ ≤

✓ λ∆ 1 + λ∆ ◆`

= exp(−Ω(`)) as long as Δ = O(1) strong spatial mixing

δ = λ Pd

i=1 xiδi

1 + λ Pd

i=1 xi

error:

δ

δi

∆i

`

connective constants:

recursion:

∆` =

d

X

i=1

∆(`−1)

i

d

initial δ’s

for hardcore model (independent sets):

f(~ x) =

d

Y

i=1

1 1 + xi

δ = Φ(f)

d

X

i=1

• ∂f

∂xi

• δi

Φ(xi)

δ

δi

choose

∆i

`

connective constants:

recursion:

∆` =

d

X

i=1

∆(`−1)

i

d

initial δ’s

for hardcore model (independent sets):

f(~ x) =

d

Y

i=1

1 1 + xi = s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi i

Φ(x) = 1 p x(1 + x)

[Li, Lu, Y. 2013]

δ = Φ(f)

d

X

i=1

• ∂f

∂xi

• δi

Φ(xi)

δ

δi

choose

∆i

`

connective constants:

recursion:

∆` =

d

X

i=1

∆(`−1)

i

d

Goal:

initial δ = O(1) arbitrary

initial δ’s

= exp(−Ω(`)) final for hardcore model (independent sets):

f(~ x) =

d

Y

i=1

1 1 + xi = s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi i

Φ(x) = 1 p x(1 + x)

~ x ∈ [0, ∞)d

λ < λc(∆) = ∆∆ (∆ − 1)(∆+1)

[Li, Lu, Y. 2013]

= s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi i

δ

∆` =

d

X

i=1

∆(`−1)

i

= s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi i

δ

∆` =

d

X

i=1

∆(`−1)

i

≤ s f(~ x) 1 + f(~ x) d X

i=1

✓ xi 1 + xi ◆ p

2 !1/p d

X

i=1

q

i

!1/q

1 p + 1 q = 1

for (Hölder’s inequality)

= s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi i

δ

∆` =

d

X

i=1

∆(`−1)

i

≤ s f(~ x) 1 + f(~ x) d X

i=1

✓ xi 1 + xi ◆ p

2 !1/p d

X

i=1

q

i

!1/q

1 p + 1 q = 1

for (Hölder’s inequality)

↵(~ x) , ✓ f(~ x) 1 + f(~ x) ◆ p

2

d

X

i=1

✓ xi 1 + xi ◆ p

2

consider

q ≤ ↵(~ x)q−1

d

X

i=1

q

i

= s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi i

δ

∆` =

d

X

i=1

∆(`−1)

i

≤ s f(~ x) 1 + f(~ x) d X

i=1

✓ xi 1 + xi ◆ p

2 !1/p d

X

i=1

q

i

!1/q

1 p + 1 q = 1

for (Hölder’s inequality)

↵(~ x) , ✓ f(~ x) 1 + f(~ x) ◆ p

2

d

X

i=1

✓ xi 1 + xi ◆ p

2

consider

λc(∆) = ∆∆ (∆ − 1)(∆+1)

is its inverse

1 p = ∆c − 1 2 ln ✓ 1 + 1 ∆c − 1 ◆

choose

∆c = ∆c(λ)

q ≤ ↵(~ x)q−1

d

X

i=1

q

i

= s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi i

δ

∆` =

d

X

i=1

∆(`−1)

i

≤ s f(~ x) 1 + f(~ x) d X

i=1

✓ xi 1 + xi ◆ p

2 !1/p d

X

i=1

q

i

!1/q

1 p + 1 q = 1

for (Hölder’s inequality)

↵(~ x) , ✓ f(~ x) 1 + f(~ x) ◆ p

2

d

X

i=1

✓ xi 1 + xi ◆ p

2

consider

λc(∆) = ∆∆ (∆ − 1)(∆+1)

is its inverse

1 p = ∆c − 1 2 ln ✓ 1 + 1 ∆c − 1 ◆

choose

∆c = ∆c(λ)

q ≤ ↵(~ x)q−1

d

X

i=1

q

i

↵(~ x) ≤ ∆c

−

1 q−1

= s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi i

δ

∆` =

d

X

i=1

∆(`−1)

i

≤ s f(~ x) 1 + f(~ x) d X

i=1

✓ xi 1 + xi ◆ p

2 !1/p d

X

i=1

q

i

!1/q

1 p + 1 q = 1

for (Hölder’s inequality)

↵(~ x) , ✓ f(~ x) 1 + f(~ x) ◆ p

2

d

X

i=1

✓ xi 1 + xi ◆ p

2

consider

λc(∆) = ∆∆ (∆ − 1)(∆+1)

is its inverse

1 p = ∆c − 1 2 ln ✓ 1 + 1 ∆c − 1 ◆

choose

∆c = ∆c(λ)

q ≤ ↵(~ x)q−1

d

X

i=1

q

i

δi ≤ ✓ ∆i ∆c ◆ `−1

q

I.H.: ↵(~ x) ≤ ∆c

−

1 q−1

= s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi i

δ

∆` =

d

X

i=1

∆(`−1)

i

≤ s f(~ x) 1 + f(~ x) d X

i=1

✓ xi 1 + xi ◆ p

2 !1/p d

X

i=1

q

i

!1/q

1 p + 1 q = 1

for (Hölder’s inequality)

↵(~ x) , ✓ f(~ x) 1 + f(~ x) ◆ p

2

d

X

i=1

✓ xi 1 + xi ◆ p

2

consider

λc(∆) = ∆∆ (∆ − 1)(∆+1)

is its inverse

1 p = ∆c − 1 2 ln ✓ 1 + 1 ∆c − 1 ◆

choose

∆c = ∆c(λ)

q ≤ ↵(~ x)q−1

d

X

i=1

q

i

δi ≤ ✓ ∆i ∆c ◆ `−1

q

I.H.:

δq ≤ 1 ∆c

d

X

i=1

✓ ∆i ∆c ◆`−1

= ✓ ∆ ∆c ◆`

↵(~ x) ≤ ∆c

−

1 q−1

= s f(~ x) 1 + f(~ x)

d

X

i=1

r xi 1 + xi i

δ

∆` =

d

X

i=1

∆(`−1)

i

≤ s f(~ x) 1 + f(~ x) d X

i=1

✓ xi 1 + xi ◆ p

2 !1/p d

X

i=1

q

i

!1/q

1 p + 1 q = 1

for (Hölder’s inequality)

↵(~ x) , ✓ f(~ x) 1 + f(~ x) ◆ p

2

d

X

i=1

✓ xi 1 + xi ◆ p

2

consider

λc(∆) = ∆∆ (∆ − 1)(∆+1)

is its inverse

1 p = ∆c − 1 2 ln ✓ 1 + 1 ∆c − 1 ◆

choose

∆c = ∆c(λ)

q ≤ ↵(~ x)q−1

d

X

i=1

q

i

δi ≤ ✓ ∆i ∆c ◆ `−1

q

I.H.:

δq ≤ 1 ∆c

d

X

i=1

✓ ∆i ∆c ◆`−1

= ✓ ∆ ∆c ◆`

= exp(−Ω(−`)) λ < λc(∆)

if

↵(~ x) ≤ ∆c

−

1 q−1

Summary

• FPTAS for monomer-dimer (counting weighted

matchings) and hardcore (counting weighted independent sets) models tightly relying on connective constants.

• Same results for sparse random graph G(n, Δ/n), greatly

improving previous results [Mossel, Sly, SODA 2007].

• Imply SSM bounds for special lattices:

Max. Previous SSM bound Connective Constant SSM Lattice degree

• ∆

T 6 0.762 [30] 4.251 419 [1] H 3 4.0 [30] 1.847 760 [4] Z2 4 2.48 [22,27] 2.679 193 [21] Z3 6 0.762 [30] 4.7387 [21] Z4 8 0.490 [30] 6.8040 [21] Z5 10 0.360 [30] 8.8602 [21] Z6 12 0.285 [30] 10.8886 [29] bound in [24] Our SSM bound

• 0.937

0.961 4.706 4.976 2.007 2.082 (2.539?) 0.816 0.822 0.506 0.508 0.367 0.367 0.288 0.288

Thank you!

Any questions?