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Random Cluster Dynamics for the Ising model is Rapidly Mixing Heng Guo Queen Mary, University of London Joint work with Mark Jerrum Oxford Nov 03 2016 Heng Guo (QMUL) Random Cluster 2016/11/03 1 / 41 The model and its dynamics Heng Guo

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Random Cluster Dynamics for the Ising model is Rapidly Mixing

Heng Guo

Queen Mary, University of London

Joint work with Mark Jerrum

Oxford Nov 03 2016

Heng Guo (QMUL) Random Cluster 2016/11/03 1 / 41

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The model and its dynamics

Heng Guo (QMUL) Random Cluster 2016/11/03 2 / 41

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The random cluster model [Fortuin, Kasteleyn 1969] Parameters 0 ⩽ p ⩽ 1 (edge weight), q ⩾ 0 (cluster weight). Given graph G = (V, E), the measure on subgraph r ⊆ E is defined as

πRC(r) ∝ p|r|(1 − p)|E\r|qκ(r),

where κ(r) is the number of connected components in (V, r).

(1 − p)4q4 p2(1 − p)2q2 p4q

Heng Guo (QMUL) Random Cluster 2016/11/03 3 / 41

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The random cluster model [Fortuin, Kasteleyn 1969] The partition function (normalizing factor): ZRC(p, q) =

∑

r⊆E

p|r|(1 − p)|E\r|qκ(r). Equivalent to the Tutte polynomial ZTutte(x, y): q = (x − 1)(y − 1) p = 1 − 1 y

Heng Guo (QMUL) Random Cluster 2016/11/03 4 / 41

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The random cluster model [Fortuin, Kasteleyn 1969]

πRC(r) ∝ p|r|(1 − p)|E\r|qκ(r)

The motivation is to unify: Ising model Potts model Bond percolation Electrical network

Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

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The random cluster model [Fortuin, Kasteleyn 1969]

πRC(r) ∝ p|r|(1 − p)|E\r|qκ(r)

The motivation is to unify: Ising model q = 2 Potts model Bond percolation Electrical network

Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

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The random cluster model [Fortuin, Kasteleyn 1969]

πRC(r) ∝ p|r|(1 − p)|E\r|qκ(r)

The motivation is to unify: Ising model q = 2 Potts model q > 2, integer Bond percolation Electrical network

Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

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The random cluster model [Fortuin, Kasteleyn 1969]

πRC(r) ∝ p|r|(1 − p)|E\r|qκ(r)

The motivation is to unify: Ising model q = 2 Potts model q > 2, integer Bond percolation q = 1 (On Kn, Erd˝

  • s-Rényi random graph)

Electrical network

Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

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The random cluster model [Fortuin, Kasteleyn 1969]

πRC(r) ∝ p|r|(1 − p)|E\r|qκ(r)

The motivation is to unify: Ising model q = 2 Potts model q > 2, integer Bond percolation q = 1 (On Kn, Erd˝

  • s-Rényi random graph)

Electrical network q → 0 (Spanning trees if p → 0 and q

p → 0)

Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

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The random cluster model [Fortuin, Kasteleyn 1969]

πRC(r) ∝ p|r|(1 − p)|E\r|qκ(r)

The motivation is to unify: Ising model q = 2 Potts model q > 2, integer Bond percolation q = 1 (On Kn, Erd˝

  • s-Rényi random graph)

Electrical network q → 0 (Spanning trees if p → 0 and q

p → 0)

Heng Guo (QMUL) Random Cluster 2016/11/03 5 / 41

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Glauber dynamics

Glauber dynamics (single edge update) PRC (Metropolis): Current state x ⊆ E

1

With prob. 1/2 do nothing. (Lazy)

2

Otherwise, choose an edge e u.a.r.

3

Move to y = x ⊕ {e} with prob. min { 1, πRC(y)

πRC(x)

} . Detailed balance: π(x)P(x, y) = π(y)P(y, x) = min{π(x), π(y)}

Heng Guo (QMUL) Random Cluster 2016/11/03 6 / 41

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Glauber dynamics

Glauber dynamics (single edge update) PRC (Metropolis): PRC(x, y) =           

1 2m min

{ 1, πRC(y)

πRC(x)

} if |x ⊕ y| = 1; 1 −

1 2m

∑

e∈E min

{ 1, πRC(x⊕{e})

πRC(x)

} if x = y;

  • therwise.

We are interested in the mixing time τϵ(PRC): τϵ(PRC) = min { t : ||Pt

RC(x0, ·) − π||TV ⩽ ϵ

} .

Heng Guo (QMUL) Random Cluster 2016/11/03 7 / 41

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A simple example

Let p < 1/2. min { 1, πRC(x ∪ {e}) πRC(x) } =     

p 1−p

if e is not a cut edge

p q(1−p)

if e is a cut edge

Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

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A simple example

Let p < 1/2. min { 1, πRC(x ∪ {e}) πRC(x) } =     

p 1−p

if e is not a cut edge

p q(1−p)

if e is a cut edge

Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

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A simple example

Let p < 1/2. min { 1, πRC(x ∪ {e}) πRC(x) } =     

p 1−p

if e is not a cut edge

p q(1−p)

if e is a cut edge

Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

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A simple example

Let p < 1/2. min { 1, πRC(x ∪ {e}) πRC(x) } =     

p 1−p

if e is not a cut edge

p q(1−p)

if e is a cut edge

Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

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A simple example

Let p < 1/2. min { 1, πRC(x ∪ {e}) πRC(x) } =     

p 1−p

if e is not a cut edge

p q(1−p)

if e is a cut edge

Heng Guo (QMUL) Random Cluster 2016/11/03 8 / 41

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Brief History

Studied extensively for special graphs, such as the complete graph (mean-field) and the lattice Z2. Mean-field: [Gore, Jerrum 1999] [Blanca, Sinclair 2015] Z2: [Borgs et al. 1999] [Blanca, Sinclair 2016] [Gheissari, Lubetzky 2016] q > 2: Slow mixing for the complete graph. 0 ⩽ q ⩽ 2: No known fast mixing bound for general graphs.

Heng Guo (QMUL) Random Cluster 2016/11/03 9 / 41

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Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2,

τϵ(PRC) ⩽ 10n4m2(ln πRC(x0)−1 + ln ϵ−1).

For q > 2, there exists p such that PRC is slow mixing on complete

  • graphs. [Gore, Jerrum 1999] [Blanca, Sinclair 2015]

For q > 2 and 0 < p < 1, it is #BIS-hard to approximate ZRC(p, q). [Goldberg, Jerrum 2012] For 0 ⩽ q < 2, there is no known obstacle.

Heng Guo (QMUL) Random Cluster 2016/11/03 10 / 41

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Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2,

τϵ(PRC) ⩽ 10n4m2(ln πRC(x0)−1 + ln ϵ−1).

For q > 2, there exists p such that PRC is slow mixing on complete

  • graphs. [Gore, Jerrum 1999] [Blanca, Sinclair 2015]

For q > 2 and 0 < p < 1, it is #BIS-hard to approximate ZRC(p, q). [Goldberg, Jerrum 2012] For 0 ⩽ q < 2, there is no known obstacle.

Heng Guo (QMUL) Random Cluster 2016/11/03 10 / 41

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Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2,

τϵ(PRC) ⩽ 10n4m2(ln πRC(x0)−1 + ln ϵ−1).

For q > 2, there exists p such that PRC is slow mixing on complete

  • graphs. [Gore, Jerrum 1999] [Blanca, Sinclair 2015]

For q > 2 and 0 < p < 1, it is #BIS-hard to approximate ZRC(p, q). [Goldberg, Jerrum 2012] For 0 ⩽ q < 2, there is no known obstacle.

Heng Guo (QMUL) Random Cluster 2016/11/03 10 / 41

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Main theorem Theorem For the random cluster model with parameters 0 < p < 1 and q = 2,

τϵ(PRC) ⩽ 10n4m2(ln πRC(x0)−1 + ln ϵ−1).

For q > 2, there exists p such that PRC is slow mixing on complete

  • graphs. [Gore, Jerrum 1999] [Blanca, Sinclair 2015]

For q > 2 and 0 < p < 1, it is #BIS-hard to approximate ZRC(p, q). [Goldberg, Jerrum 2012] For 0 ⩽ q < 2, there is no known obstacle.

Heng Guo (QMUL) Random Cluster 2016/11/03 10 / 41

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Swandsen-Wang algorithm

Heng Guo (QMUL) Random Cluster 2016/11/03 11 / 41

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Ferromagnetic Ising model [Ising, Lenz 1925]

Parameter β > 1. A configuration σ : V → {+, −}. πIsing(σ) ∝ βmono(σ) = βm−cut(σ) Partition function ZIsing(β) = ∑

σ βmono(σ) β0 β2 β4

Heng Guo (QMUL) Random Cluster 2016/11/03 12 / 41

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Equivalence at q = 2 Let β =

1 1−p.

ZIsing(β) = β|E|ZRC (p, 2)

Heng Guo (QMUL) Random Cluster 2016/11/03 13 / 41

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Swendsen-Wang algorithm [Swendsen, Wang 1987]

A global Markov chain to sample Ising configurations. Current configuration σ

1

Mark all monochromatic edges under σ as M

2

Remove each edge in M with probability β−1 (Recall β−1 = 1 − p)

3

Assign a random spin to each component of (V, M) Practically very fast for the Ising model, but difficult to analyze. Conjectured to be rapidly mixing for all graphs. (Open problem since 90s.)

Heng Guo (QMUL) Random Cluster 2016/11/03 14 / 41

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Another simple example

1

Activate mono edges

2

Re-randomize mono edges

3

Color components

Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

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Another simple example

1

Activate mono edges

2

Re-randomize mono edges

3

Color components

Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

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Another simple example

1

Activate mono edges

2

Re-randomize mono edges

3

Color components

Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

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Another simple example

1

Activate mono edges

2

Re-randomize mono edges

3

Color components

Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

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Another simple example

1

Activate mono edges

2

Re-randomize mono edges

3

Color components

Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

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Another simple example

1

Activate mono edges

2

Re-randomize mono edges

3

Color components

Heng Guo (QMUL) Random Cluster 2016/11/03 15 / 41

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Previous Results Swendsen-Wang algorithm on the complete graph: [Gore, Jerrum 1999] [Cooper, Dyer, Frieze, Rue 2000] [Long, Nachimus, Ning, Peres 2011] [Borgs, Chayes, Tetali 2011] [Galanis, Štefankoviˇ c, Vigoda 2015] Theorem (Ullrich 2014)

τϵ(PSW) ⩽ τϵ(PRC)

Heng Guo (QMUL) Random Cluster 2016/11/03 16 / 41

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Previous Results Swendsen-Wang algorithm on the complete graph: [Gore, Jerrum 1999] [Cooper, Dyer, Frieze, Rue 2000] [Long, Nachimus, Ning, Peres 2011] [Borgs, Chayes, Tetali 2011] [Galanis, Štefankoviˇ c, Vigoda 2015] Theorem (Ullrich 2014)

τϵ(PSW) ⩽ τϵ(PRC)

Heng Guo (QMUL) Random Cluster 2016/11/03 16 / 41

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Concequence — Swendsen-Wang algorithm is rapidly mixing Theorem (Ullrich 2014)

τϵ(PSW) ⩽ τϵ(PRC)

Combine with our theorem: the Swendsen-Wang algorithm is rapidly mixing at q = 2, namely, for the ferromagnetic Ising model at any temperature.

The Swendsen-Wang algorithm is conjectured to have a n1/4 mixing time (by Peres and Sokal).

Heng Guo (QMUL) Random Cluster 2016/11/03 17 / 41

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Concequence — Swendsen-Wang algorithm is rapidly mixing Theorem (Ullrich 2014)

τϵ(PSW) ⩽ τϵ(PRC)

Combine with our theorem: the Swendsen-Wang algorithm is rapidly mixing at q = 2, namely, for the ferromagnetic Ising model at any temperature.

The Swendsen-Wang algorithm is conjectured to have a n1/4 mixing time (by Peres and Sokal).

Heng Guo (QMUL) Random Cluster 2016/11/03 17 / 41

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Even subgraphs

Heng Guo (QMUL) Random Cluster 2016/11/03 18 / 41

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Another equivalent formulations at q = 2

Even subgraphs

Let r ⊆ E such that every vertex in (V, r) has an even degree. πeven(r) ∝ p|r|(1 − p)|E\r| Partition function Zeven(p)

(1 − p)4 NOT EVEN p4

Heng Guo (QMUL) Random Cluster 2016/11/03 19 / 41

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Equivalence at q = 2 Let β =

1 1−p.

ZIsing(β) = β|E|ZRC (p, 2) = 2|V|β|E|Zeven

(p

2

)

Heng Guo (QMUL) Random Cluster 2016/11/03 20 / 41

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Equivalence at q = 2

Random-cluster (p, 2) Ising model β = (1 − p)−1 Even subgraphs p/2 [Edwards, Sokal 1988] [Fortuin, Kasteleyn 1969] [Grimmett, Janson 2009] [van der Waerden 1941] Slow mixing FPRAS [Jerrum, Sinclair 93] This talk

Heng Guo (QMUL) Random Cluster 2016/11/03 21 / 41

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Equivalence at q = 2

Random-cluster (p, 2) Ising model β = (1 − p)−1 Even subgraphs p/2 [Edwards, Sokal 1988] [Fortuin, Kasteleyn 1969] [Grimmett, Janson 2009] [van der Waerden 1941] Slow mixing FPRAS [Jerrum, Sinclair 93] This talk

Heng Guo (QMUL) Random Cluster 2016/11/03 21 / 41

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Equivalence at q = 2

Random-cluster (p, 2) Ising model β = (1 − p)−1 Even subgraphs p/2 [Edwards, Sokal 1988] [Fortuin, Kasteleyn 1969] [Grimmett, Janson 2009] [van der Waerden 1941] Slow mixing FPRAS [Jerrum, Sinclair 93] This talk

Heng Guo (QMUL) Random Cluster 2016/11/03 21 / 41

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Equivalence at q = 2

Random-cluster (p, 2) Ising model β = (1 − p)−1 Even subgraphs p/2 [Edwards, Sokal 1988] [Fortuin, Kasteleyn 1969] [Grimmett, Janson 2009] [van der Waerden 1941] Slow mixing FPRAS [Jerrum, Sinclair 93] This talk

Heng Guo (QMUL) Random Cluster 2016/11/03 21 / 41

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Equivalence at q = 2

Random-cluster (p, 2) Ising model β = (1 − p)−1 Even subgraphs p/2 [Edwards, Sokal 1988] [Fortuin, Kasteleyn 1969] [Grimmett, Janson 2009] [van der Waerden 1941] Slow mixing FPRAS [Jerrum, Sinclair 93] This talk

Heng Guo (QMUL) Random Cluster 2016/11/03 21 / 41

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Equivalence at q = 2

Random-cluster (p, 2) Ising model β = (1 − p)−1 Even subgraphs p/2 [Edwards, Sokal 1988] [Fortuin, Kasteleyn 1969] [Grimmett, Janson 2009] [van der Waerden 1941] Slow mixing FPRAS [Jerrum, Sinclair 93] This talk

Heng Guo (QMUL) Random Cluster 2016/11/03 21 / 41

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Equivalence at q = 2

Random-cluster (p, 2) Ising model β = (1 − p)−1 Even subgraphs p/2 [Edwards, Sokal 1988] [Fortuin, Kasteleyn 1969] [Grimmett, Janson 2009] [van der Waerden 1941] Slow mixing FPRAS [Jerrum, Sinclair 93] This talk

Heng Guo (QMUL) Random Cluster 2016/11/03 21 / 41

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Grimmett-Janson coupling Given a graph G, draw a random even subgraph S ⊆ E with p ⩽ 1

2:

Pr(S = s) = πeven(s). Then we add every edge e ̸∈ S with probability p′ =

p 1−p.

Call this subgraph R. Theorem (Grimmett, Janson 2009) Pr(R = r) = πRC; 2p,2(r).

Heng Guo (QMUL) Random Cluster 2016/11/03 22 / 41

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Grimmett-Janson coupling Given a graph G, draw a random even subgraph S ⊆ E with p ⩽ 1

2:

Pr(S = s) = πeven(s). Then we add every edge e ̸∈ S with probability p′ =

p 1−p.

Call this subgraph R. Theorem (Grimmett, Janson 2009) Pr(R = r) = πRC; 2p,2(r).

Heng Guo (QMUL) Random Cluster 2016/11/03 22 / 41

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The Proof

Heng Guo (QMUL) Random Cluster 2016/11/03 23 / 41

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Bound the mixing time

A Markov chain is a random walk on its state space (exponentially large).

↔ ↔

Heng Guo (QMUL) Random Cluster 2016/11/03 24 / 41

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Bound the mixing time

A Markov chain is a random walk on its state space (exponentially large).

↔ ↔

▶ There are 2|E| many configurations. ▶ Two configurations are adjacent if they differ by exactly one edge. Heng Guo (QMUL) Random Cluster 2016/11/03 24 / 41

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Bound the mixing time

A Markov chain is a random walk on its state space (exponentially large).

↔ ↔

▶ There are 2|E| many configurations. ▶ Two configurations are adjacent if they differ by exactly one edge.

Rapidly mixing ⇔ The state space is very well connected.

Heng Guo (QMUL) Random Cluster 2016/11/03 24 / 41

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Congestion and canonical paths

Construct a set Γ of canonical paths γxy for all pairs of states (x, y). The key quantity of Γ is its congestion: ρ(Γ) := max

(z,z ′)∈Ω2 P(z,z ′)>0

L π(z)P(z, z ′) ∑

x,y∈Ω2 γxy∋(z,z ′)

w(γxy), where w(γxy) = π(x)π(y). Theorem (Sinclair 1992) τε(P) ⩽ ρ(Γ)(ln π(x0)−1 + ln ε−1).

Heng Guo (QMUL) Random Cluster 2016/11/03 25 / 41

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Alternative view of canonical paths Fix Γ = {γxy} and an integer k ⩽ L.

1

Draw the initial and final states I and F independently according to π(·).

2

A random path γIF ∈ Γ. µ(γIF) = w(γIF) = π(I)π(F)

3

Let Zk be the kth state of γIF. (Assume all paths in Γ have the same length L.)

The congestion ρ(Γ) is polynomial related with maxk

Pr(Zk=z) π(z)

.

Heng Guo (QMUL) Random Cluster 2016/11/03 26 / 41

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Alternative view in action Let q = 1. Then πRC(·) is a product measure.

G:

e1 e2 e3 e4 Heng Guo (QMUL) Random Cluster 2016/11/03 27 / 41

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Alternative view in action Let q = 1. Then πRC(·) is a product measure.

G:

e1 e2 e3 e4

I F

Heng Guo (QMUL) Random Cluster 2016/11/03 27 / 41

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SLIDE 57

Alternative view in action Let q = 1. Then πRC(·) is a product measure.

G:

e1 e2 e3 e4

σF σI σF σI Z2: I F

Heng Guo (QMUL) Random Cluster 2016/11/03 27 / 41

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SLIDE 58

Alternative view in action Let q = 1. Then πRC(·) is a product measure.

G:

e1 e2 e3 e4

σF σI σF σI Z2: I F

Heng Guo (QMUL) Random Cluster 2016/11/03 27 / 41

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SLIDE 59

Alternative view in action Let q = 1. Then πRC(·) is a product measure.

G:

e1 e2 e3 e4

σF σI σF σI Z2: I F

Heng Guo (QMUL) Random Cluster 2016/11/03 27 / 41

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SLIDE 60

Alternative view in action Let q = 1. Then πRC(·) is a product measure.

G:

e1 e2 e3 e4

σF σI σF σI Z2: I F

Pr(Zk = z)

π(z) = 1

Heng Guo (QMUL) Random Cluster 2016/11/03 27 / 41

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SLIDE 61

From paths to flows

Instead of one path from x to y, we can have a random path from x to y. Flow Γ is a collection of paths equipped with weights w(·) such that ∑

γ is from x to y

w(γ) = π(x)π(y). Zk is defined similarly.

1

Random initial and final states I and F

2

A random path γ from I to F according to w(·).

3

Zk is the kth state of γ. We will look at Pr(Zk=z)

π(z)

.

Heng Guo (QMUL) Random Cluster 2016/11/03 28 / 41

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SLIDE 62

Lifting canonical paths In an ideal world . . .

Suppose we have canonical paths Γeven for even subgraphs with low

  • congestion. (similar to [Jerrum, Sinclair 93])

Then use Grimmett-Janson to lift Γeven to a flow for random cluster.

I = W0 W1 W2 WL−1 WL = F Z0 Z1 Z2 ZL−1 ZL

Grimmett-Janson Grimmett-Janson Grimmett-Janson Grimmett-Janson Grimmett-Janson

w(ζ) = w(γ) Pr(γ → ζ)

Heng Guo (QMUL) Random Cluster 2016/11/03 29 / 41

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Ideal lifting

If Wk deviates from πeven(·) by at most polynomial, then so does Zk from πRC(·). Pr(Wk = w) πeven(w) ⩽ nO(1)ρ(Γ) Pr(Zk = z) = ∑

w⊆z, w even

Pr(Wk = w) ( p 1 − p )|z\w| (1 − 2p 1 − p )|E\z| ⩽ nO(1)ρ(Γ) ∑

w⊆z, w even

πeven(w) ( p 1 − p )|z\w| (1 − 2p 1 − p )|E\z| = nO(1)ρ(Γ)πRC(z) (by GJ)

Heng Guo (QMUL) Random Cluster 2016/11/03 30 / 41

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SLIDE 64

Ideal lifting

If Wk deviates from πeven(·) by at most polynomial, then so does Zk from πRC(·). Pr(Wk = w) πeven(w) ⩽ nO(1)ρ(Γ) Pr(Zk = z) = ∑

w⊆z, w even

Pr(Wk = w) ( p 1 − p )|z\w| (1 − 2p 1 − p )|E\z| ⩽ nO(1)ρ(Γ) ∑

w⊆z, w even

πeven(w) ( p 1 − p )|z\w| (1 − 2p 1 − p )|E\z| = nO(1)ρ(Γ)πRC(z) (by GJ)

Heng Guo (QMUL) Random Cluster 2016/11/03 30 / 41

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In the real world . . . Two issues:

1

We do not have good canonical paths for even subgraphs — Jerrum-Sinclair chain moves among all subgraphs!

2

Grimmett-Janson adds indepdendent edges — Zi and Zi+1 are not adjacent states! They may differ by a lot of edges.

Heng Guo (QMUL) Random Cluster 2016/11/03 31 / 41

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SLIDE 66

In the real world . . . Two issues:

1

We do not have good canonical paths for even subgraphs — Jerrum-Sinclair chain moves among all subgraphs!

2

Grimmett-Janson adds indepdendent edges — Zi and Zi+1 are not adjacent states! They may differ by a lot of edges.

Heng Guo (QMUL) Random Cluster 2016/11/03 31 / 41

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Patch 1 Issue 1: need canonical paths for even subgraphs.

Construct paths Γeven = {γxy} where x and y are both even subgraphs.

▶ x ⊕ y is also even.

x ⊕ y can be covered by edge-disjoint cycles.

▶ Pick a canonical ordering of edges. Unwind each cycle:

W0 = x, Wi = Wi−1 ⊕ ei

▶ Enlarge the state space to all even and near-even subgraphs.

Every path is in the augmented space. Γeven has low congestion — same reason as [Jerrum, Sinclair 1993].

Heng Guo (QMUL) Random Cluster 2016/11/03 32 / 41

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Patch 1 Issue 1: need canonical paths for even subgraphs.

Construct paths Γeven = {γxy} where x and y are both even subgraphs.

▶ x ⊕ y is also even.

x ⊕ y can be covered by edge-disjoint cycles.

▶ Pick a canonical ordering of edges. Unwind each cycle:

W0 = x, Wi = Wi−1 ⊕ ei

▶ Enlarge the state space to all even and near-even subgraphs.

Every path is in the augmented space. Γeven has low congestion — same reason as [Jerrum, Sinclair 1993].

Heng Guo (QMUL) Random Cluster 2016/11/03 32 / 41

slide-69
SLIDE 69

Patch 1 Issue 1: need canonical paths for even subgraphs.

Construct paths Γeven = {γxy} where x and y are both even subgraphs.

▶ x ⊕ y is also even.

x ⊕ y can be covered by edge-disjoint cycles.

▶ Pick a canonical ordering of edges. Unwind each cycle:

W0 = x, Wi = Wi−1 ⊕ ei

▶ Enlarge the state space to all even and near-even subgraphs.

Every path is in the augmented space. Γeven has low congestion — same reason as [Jerrum, Sinclair 1993].

Heng Guo (QMUL) Random Cluster 2016/11/03 32 / 41

slide-70
SLIDE 70

Patch 1 Issue 1: need canonical paths for even subgraphs.

Construct paths Γeven = {γxy} where x and y are both even subgraphs.

▶ x ⊕ y is also even.

x ⊕ y can be covered by edge-disjoint cycles.

▶ Pick a canonical ordering of edges. Unwind each cycle:

W0 = x, Wi = Wi−1 ⊕ ei

▶ Enlarge the state space to all even and near-even subgraphs.

Every path is in the augmented space. Γeven has low congestion — same reason as [Jerrum, Sinclair 1993].

Heng Guo (QMUL) Random Cluster 2016/11/03 32 / 41

slide-71
SLIDE 71

Patch 1 Issue 1: need canonical paths for even subgraphs.

Construct paths Γeven = {γxy} where x and y are both even subgraphs.

▶ x ⊕ y is also even.

x ⊕ y can be covered by edge-disjoint cycles.

▶ Pick a canonical ordering of edges. Unwind each cycle:

W0 = x, Wi = Wi−1 ⊕ ei

▶ Enlarge the state space to all even and near-even subgraphs.

Every path is in the augmented space. Γeven has low congestion — same reason as [Jerrum, Sinclair 1993].

Heng Guo (QMUL) Random Cluster 2016/11/03 32 / 41

slide-72
SLIDE 72

Patch 1 Issue 1: need canonical paths for even subgraphs.

x = Z0 y = Z6 x ⊕ y

Heng Guo (QMUL) Random Cluster 2016/11/03 33 / 41

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SLIDE 73

Patch 1 Issue 1: need canonical paths for even subgraphs.

x = Z0 y = Z6 x ⊕ y Z1

Heng Guo (QMUL) Random Cluster 2016/11/03 33 / 41

slide-74
SLIDE 74

Patch 1 Issue 1: need canonical paths for even subgraphs.

x = Z0 y = Z6 x ⊕ y Z1 Z2

Heng Guo (QMUL) Random Cluster 2016/11/03 33 / 41

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SLIDE 75

Patch 1 Issue 1: need canonical paths for even subgraphs.

x = Z0 y = Z6 x ⊕ y Z1 Z2 Z3

Heng Guo (QMUL) Random Cluster 2016/11/03 33 / 41

slide-76
SLIDE 76

Patch 1 Issue 1: need canonical paths for even subgraphs.

x = Z0 y = Z6 x ⊕ y Z1 Z2 Z3 Z4

Heng Guo (QMUL) Random Cluster 2016/11/03 33 / 41

slide-77
SLIDE 77

Patch 1 Issue 1: need canonical paths for even subgraphs.

x = Z0 y = Z6 x ⊕ y Z1 Z2 Z3 Z4 Z5

Heng Guo (QMUL) Random Cluster 2016/11/03 33 / 41

slide-78
SLIDE 78

Patch 1 Issue 1: need canonical paths for even subgraphs.

x = Z0 y = Z6 x ⊕ y Z1 Z2 Z3 Z4 Z5

Heng Guo (QMUL) Random Cluster 2016/11/03 33 / 41

slide-79
SLIDE 79

Patch 1 Issue 1: need canonical paths for even subgraphs.

x = Z0 y = Z6 x ⊕ y Z1 Z2 Z3 Z4 Z5

Heng Guo (QMUL) Random Cluster 2016/11/03 33 / 41

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SLIDE 80

Patch 1 Issue 1: need canonical paths for even subgraphs. Γeven has low congestion — combinatorial encoding [Jerrum, Sinclair 1993].

For any γxy ∋ (z, z ′), let u = x ⊕ y ⊕ z. This mapping is injective.

x y z z ′

Heng Guo (QMUL) Random Cluster 2016/11/03 34 / 41

slide-81
SLIDE 81

Patch 1 Issue 1: need canonical paths for even subgraphs. Γeven has low congestion — combinatorial encoding [Jerrum, Sinclair 1993].

For any γxy ∋ (z, z ′), let u = x ⊕ y ⊕ z. This mapping is injective.

x y z z ′ x ⊕ y

Heng Guo (QMUL) Random Cluster 2016/11/03 34 / 41

slide-82
SLIDE 82

Patch 1 Issue 1: need canonical paths for even subgraphs. Γeven has low congestion — combinatorial encoding [Jerrum, Sinclair 1993].

For any γxy ∋ (z, z ′), let u = x ⊕ y ⊕ z. This mapping is injective.

x y z z ′ x ⊕ y u

Heng Guo (QMUL) Random Cluster 2016/11/03 34 / 41

slide-83
SLIDE 83

Patch 1 Issue 1: need canonical paths for even subgraphs. Γeven has low congestion — combinatorial encoding [Jerrum, Sinclair 1993].

For any γxy ∋ (z, z ′), let u = x ⊕ y ⊕ z. This mapping is injective.

x y z z ′ x ⊕ y u

π(x)π(y) = π(z)π(u)

Heng Guo (QMUL) Random Cluster 2016/11/03 34 / 41

slide-84
SLIDE 84

Patch 1 Issue 1: need canonical paths for even subgraphs. Γeven has low congestion — combinatorial encoding [Jerrum, Sinclair 1993].

For any γxy ∋ (z, z ′), let u = x ⊕ y ⊕ z. This mapping is injective.

x y z z ′ x ⊕ y u

∑

γxy∋(z,z ′)

π(x)π(y) ⩽ π(z) ∑

u

π(u)

Heng Guo (QMUL) Random Cluster 2016/11/03 34 / 41

slide-85
SLIDE 85

Patch 1 Issue 1: need canonical paths for even subgraphs. Γeven has low congestion — combinatorial encoding [Jerrum, Sinclair 1993].

For any γxy ∋ (z, z ′), let u = x ⊕ y ⊕ z. This mapping is injective.

x y z z ′ x ⊕ y u

∑

γxy∋(z,z ′)

π(x)π(y) ⩽ π(z) ∑

u

π(u) ⩽ π(z)

Heng Guo (QMUL) Random Cluster 2016/11/03 34 / 41

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SLIDE 86

Patch 1 Issue 1: need canonical paths for even subgraphs.

One final problem for issue 1: W0 and WL are both even, but intermediate Wi’s can be near-even. A generalization of Grimmett-Janson: Give each near-even subgraph a penalty of 1/n2. Add independent edges with prob.

p 1−p as before.

Call the resulting measure π(·).

  • π(x)

πRC(x) = Θ(1).

Heng Guo (QMUL) Random Cluster 2016/11/03 35 / 41

slide-87
SLIDE 87

Patch 1 Issue 1: need canonical paths for even subgraphs.

One final problem for issue 1: W0 and WL are both even, but intermediate Wi’s can be near-even. A generalization of Grimmett-Janson: Give each near-even subgraph a penalty of 1/n2. Add independent edges with prob.

p 1−p as before.

Call the resulting measure π(·).

  • π(x)

πRC(x) = Θ(1).

Heng Guo (QMUL) Random Cluster 2016/11/03 35 / 41

slide-88
SLIDE 88

Patch 2 Issue 2: Zi and Zi+1 differ by more than 1 edge.

An easy fix: insert intermediate states to change edges one by one in Zi ⊕ Zi+1, which has a product measure on E\(Wi ∪ Wi+1).

Wi Wi+1 Z 0

i

Z 1

i

Z m−1

i

Z m

i

Zi = = Zi+1

Grimmett-Janson Grimmett-Janson

The distribution of Z j

i is the same as that of Zi (j < m).

Total length is mL.

Heng Guo (QMUL) Random Cluster 2016/11/03 36 / 41

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SLIDE 89

Better patch 2 Issue 2: Zi and Zi+1 differ by more than 1 edge. Lift Wi+1 to Zi+1 conditional on Zi such that Zi+1 and Zi are adjacent and the marginal of Zi+1 is correct. The marginal distributions of Z0 and ZL are correct, but their joint distribution is not — Z0 and ZL are correlated. Append a tail on the path after ZL to re-randomize edges that are not in WL. This removes the correlation. Total length is at most L + m.

Heng Guo (QMUL) Random Cluster 2016/11/03 37 / 41

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SLIDE 90

Putting everything together

W0 W1 W2 WL Z0 Z1 Z2 ZL ZL+m

Re-randomization Grimmett-Janson Grimmett-Janson

Heng Guo (QMUL) Random Cluster 2016/11/03 38 / 41

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SLIDE 91

Putting everything together

W0 W1 W2 WL Z0 Z1 Z2 ZL ZL+m

Re-randomization Grimmett-Janson Grimmett-Janson

Heng Guo (QMUL) Random Cluster 2016/11/03 38 / 41

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SLIDE 92

Putting everything together

W0 W1 W2 WL Z0 Z1 Z2 ZL ZL+m

Re-randomization Grimmett-Janson Grimmett-Janson

Heng Guo (QMUL) Random Cluster 2016/11/03 38 / 41

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SLIDE 93

Putting everything together

W0 W1 W2 WL Z0 Z1 Z2 ZL ZL+m

Re-randomization Grimmett-Janson Grimmett-Janson

W1 = W0 ∪ {e} ⇒ Z1 = Z0 ∪ {e}

Heng Guo (QMUL) Random Cluster 2016/11/03 38 / 41

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SLIDE 94

Putting everything together

W0 W1 W2 WL Z0 Z1 Z2 ZL ZL+m

Re-randomization Grimmett-Janson Grimmett-Janson

W1 = W0 ∪ {e} ⇒ Z1 = Z0 ∪ {e} W2 = W1\{e′} ⇒ Z2 =      Z1

  • prob. p′

Z1\{e′}

  • prob. 1 − p′

Heng Guo (QMUL) Random Cluster 2016/11/03 38 / 41

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SLIDE 95

Putting everything together

W0 W1 W2 WL Z0 Z1 Z2 ZL ZL+m

Re-randomization Grimmett-Janson Grimmett-Janson

W1 = W0 ∪ {e} ⇒ Z1 = Z0 ∪ {e} W2 = W1\{e′} ⇒ Z2 =      Z1

  • prob. p′

Z1\{e′}

  • prob. 1 − p′

Heng Guo (QMUL) Random Cluster 2016/11/03 38 / 41

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SLIDE 96

Putting everything together

W0 W1 W2 WL Z0 Z1 Z2 ZL ZL+m

Re-randomization Grimmett-Janson Grimmett-Janson

W1 = W0 ∪ {e} ⇒ Z1 = Z0 ∪ {e} W2 = W1\{e′} ⇒ Z2 =      Z1

  • prob. p′

Z1\{e′}

  • prob. 1 − p′

Heng Guo (QMUL) Random Cluster 2016/11/03 38 / 41

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SLIDE 97

Future directions

Heng Guo (QMUL) Random Cluster 2016/11/03 39 / 41

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SLIDE 98

Tutte polynomial [Goldberg, Jerrum 08,12,14]

q = (x −1)(y −1) Tractable

FPRAS NP-hard

(most #P-hard)

#PM-equivalent #BIS-hard Open:

All white 0 ⩽ q < 1 1 < q < 2

q = 0 q = 1 q = 2 q = 2 q = 1

x y

(1, 1) −1 −1 Heng Guo (QMUL) Random Cluster 2016/11/03 40 / 41

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SLIDE 99

Recap Theorem At q = 2, τϵ(PRC) ⩽ 10n4m2(ln πRC(x0)−1 + ln ϵ−1). q = 2 tighter mixing time bound? 1 < q < 2 (monotone) fast mixing? 0 ⩽ q < 1 (e.g. Tutte(2,1) = #Forests) fast mixing???

Heng Guo (QMUL) Random Cluster 2016/11/03 41 / 41

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SLIDE 100

Recap Theorem At q = 2, τϵ(PRC) ⩽ 10n4m2(ln πRC(x0)−1 + ln ϵ−1). q = 2 tighter mixing time bound? 1 < q < 2 (monotone) fast mixing? 0 ⩽ q < 1 (e.g. Tutte(2,1) = #Forests) fast mixing???

Thank You!

Paper available: arxiv.org/abs/1605.00139

Heng Guo (QMUL) Random Cluster 2016/11/03 41 / 41