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A Tutorial on Various Aspects of Mixing

Prasad Tetali

School of Mathematics & School of Computer Science Georgia Institute of Technology

January 9, 2009

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Acknowledgement

The part Colorings on Trees of this presentation was given to me by Juan Vera. The part on Broadcasting on a Tree was given by Nayantara Bhatnagar. I am grateful for their help.

Prasad Tetali

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Catalan Shuffling: an open question

Consider the setCn of triangulations of a convexn-sided polygon, for integern≥4.

Well-known: |Cn|= [1/(n+ 1)] 2nn

, thenth Catalan number.

I A shuffle consists of picking one of then−3 diagonals uniformly at random and “flipping the diagonal”–

removing the diagonal and in the quadrilateral created by this removal, placing the (only possible) other diagonal.

I How long until the triangulation is random?

I Conjecture (Aldous). O(n3/2) shuffles ought to be enough!,

I Best known. Lower bound : Ω(n3/2) (Mol-Ree-Ste ’99) Upper bound ofO(n4(logn)) (McS–T. ’99).

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Catalan Shuffling: an open question

Consider the setCn of triangulations of a convexn-sided polygon, for integern≥4.

Well-known: |Cn|= [1/(n+ 1)] 2nn

, thenth Catalan number.

I A shuffle consists of picking one of then−3 diagonals uniformly at random and “flipping the diagonal”–

removing the diagonal and in the quadrilateral created by this removal, placing the (only possible) other diagonal.

I How long until the triangulation is random?

I Conjecture (Aldous). O(n3/2) shuffles ought to be enough!,

I Best known. Lower bound : Ω(n3/2) (Mol-Ree-Ste ’99) Upper bound ofO(n4(logn)) (McS–T. ’99).

(5)

Catalan Shuffling: an open question

Consider the setCn of triangulations of a convexn-sided polygon, for integern≥4.

Well-known: |Cn|= [1/(n+ 1)] 2nn

, thenth Catalan number.

I A shuffle consists of picking one of then−3 diagonals uniformly at random and “flipping the diagonal”–

removing the diagonal and in the quadrilateral created by this removal, placing the (only possible) other diagonal.

I How long until the triangulation israndom?

I Conjecture (Aldous). O(n3/2) shuffles ought to be enough!,

I Best known. Lower bound : Ω(n3/2) (Mol-Ree-Ste ’99) Upper bound ofO(n4(logn)) (McS–T. ’99).

3/42

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Catalan Shuffling: an open question

Consider the setCn of triangulations of a convexn-sided polygon, for integern≥4.

Well-known: |Cn|= [1/(n+ 1)] 2nn

, thenth Catalan number.

I A shuffle consists of picking one of then−3 diagonals uniformly at random and “flipping the diagonal”–

removing the diagonal and in the quadrilateral created by this removal, placing the (only possible) other diagonal.

I How long until the triangulation israndom?

I Conjecture(Aldous). O(n3/2) shuffles ought to be enough!,

I Best known. Lower bound : Ω(n3/2) (Mol-Ree-Ste ’99) Upper bound ofO(n4(logn)) (McS–T. ’99).

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Catalan Shuffling: an open question

Consider the setCn of triangulations of a convexn-sided polygon, for integern≥4.

Well-known: |Cn|= [1/(n+ 1)] 2nn

, thenth Catalan number.

I A shuffle consists of picking one of then−3 diagonals uniformly at random and “flipping the diagonal”–

removing the diagonal and in the quadrilateral created by this removal, placing the (only possible) other diagonal.

I How long until the triangulation israndom?

I Conjecture(Aldous). O(n3/2) shuffles ought to be enough!,

I Best known. Lower bound : Ω(n3/2) (Mol-Ree-Ste ’99) Upper bound ofO(n4(logn)) (McS–T. ’99).

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The rho (nonreversible) walk – recently settled question

Consider thesimple random walkon the discrete circle Zp (for p: prime) – move fromatoa±1, with equal probability.

Well-known: O(n2) steps to get to a randomi∈ {0,1, . . . , p}.

I Add a doublingmove – from atoa±1 or 2a, with probability 1/3 independently.

I Now how long until random?

Theorem(Kim-Mon–T. ’07). O(logn log logn) moves sufficient. Motivation: Arises in the analysis of Pollard’s Rho algorithm for finding thediscrete logarithm in a cyclic group of prime order.

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The rho (nonreversible) walk – recently settled question

Consider thesimple random walkon the discrete circle Zp (for p: prime) – move fromatoa±1, with equal probability.

Well-known: O(n2) steps to get to a randomi∈ {0,1, . . . , p}.

I Add a doublingmove – from atoa±1 or 2a, with probability 1/3 independently.

I Now how long until random?

Theorem(Kim-Mon–T. ’07). O(logn log logn) moves sufficient. Motivation: Arises in the analysis of Pollard’s Rho algorithm for finding thediscrete logarithm in a cyclic group of prime order.

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The rho (nonreversible) walk – recently settled question

Consider thesimple random walkon the discrete circle Zp (for p: prime) – move fromatoa±1, with equal probability.

Well-known: O(n2) steps to get to a randomi∈ {0,1, . . . , p}.

I Add a doublingmove – from atoa±1 or 2a, with probability 1/3 independently.

I Now how long until random?

Theorem(Kim-Mon–T. ’07). O(logn log logn) moves sufficient. Motivation: Arises in the analysis of Pollard’s Rho algorithm for finding thediscrete logarithm in a cyclic group of prime order.

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The rho (nonreversible) walk – recently settled question

Consider thesimple random walkon the discrete circle Zp (for p: prime) – move fromatoa±1, with equal probability.

Well-known: O(n2) steps to get to a randomi∈ {0,1, . . . , p}.

I Add a doublingmove – from atoa±1 or 2a, with probability 1/3 independently.

I Now how long until random?

Theorem(Kim-Mon–T. ’07). O(logn log logn) moves sufficient.

Motivation: Arises in the analysis of Pollard’s Rho algorithm for finding thediscrete logarithm in a cyclic group of prime order.

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The rho (nonreversible) walk – recently settled question

Consider thesimple random walkon the discrete circle Zp (for p: prime) – move fromatoa±1, with equal probability.

Well-known: O(n2) steps to get to a randomi∈ {0,1, . . . , p}.

I Add a doublingmove – from atoa±1 or 2a, with probability 1/3 independently.

I Now how long until random?

Theorem(Kim-Mon–T. ’07). O(logn log logn) moves sufficient.

Motivation: Arises in the analysis of Pollard’s Rho algorithm for finding thediscrete logarithm in a cyclic group of prime order.

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Welcome to Markov Chains

Let (Ω, P, π) : denote a Markov chain on state space Ω, with the transition probab. matrixP, and (unique) invariant measureπ:

P(x, y)≥0,∀x, y∈Ω and X

y

P(x, y) = 1,∀x∈Ω,

X

x∈Ω

π(x)P(x, y) =π(y), ∀y∈Ω.

I Always assumeirreducible, withπ having full support on Ω.

I Typically assume aperiodic, soPn(x,·)→π(·), as n→ ∞.

I Often have reversibility :

π(x)P(x, y) =π(y)P(y, x), ∀x, y∈Ω.

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Markov Chain Monte Carlo

Typically given a LARGE set Ω, the goal is to sample elements from Ω at random from a desired distributionπ over Ω.

Usuallyπ is given implicitly.

Examples: weighted matchings, uniform independent sets, colorings etc. of a given graph.

The MCMC approach to sampling is to construct a Markov chain (based on “local” moves) on Ω which converges toπ.

I Start at any x0 ∈Ω; run the chain for T steps, and output xT.

I key question: How large doesT have to be so that the distribution of xT is a good approximation toπ?

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Some Successes

Defn. Fully poly. randomized approx. scheme(FPRAS): Given inputx, error parameter >0, and a confidence parameter δ >0, an approximation algorithm outputting A(x) (for the true valuef(x)), so that

Pr

(1−)f(x)≤A(x)≤(1 +)f(x)

≥1−δ , in time polynomial in|x|,−1 and log(1/δ).

Some examples having an FPRAS using the MCMC approach:

I Counting Matchings in General Graphs, Perfect Matchings in Bipartite Graphs, Computing the permanent of a matrix

I Counting the number of linear extensions of a poset

I Computing the volume of an n-dim. convex body

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Some Frustrations

Some notorious open problems:

I Generating perfectmatchings u.a.r. from a general G.

I Generating acyclic orientations of Gu.a.r. from a given undirected graph G

I Contingency Table Problems: given row and column sums, generate u.a.r. a table of nonneg. integers with those sums

I Is the basis-exchange walk on bases of an arbitrary matroid rapidly mixing? (known for spanning trees, and balanced matroids [F-M ’9?])

I Generating independent sets from a given bipartitegraph

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Many Paths to Mixing – Distance to Equilibrium

Letknx(y) =Pn(x, y)/π(y) density w.r.t. π at time n≥0, so under aperiodicity,kxn(y)→1.

Defn. Lp-distance:

kkn−1kpp,π =P

y∈Ω|kn(y)−1|pπ(y), 1≤p <+∞.

Forµ: probab. distribution on Ω,

I p= 1:

kµ−πktv= 1 2

X

y

|µ(y)−π(y)|= 1

2kµ/π−1k1,π,

I p= 2:

Varπ(µ/π) =X

y

µ(y) π(y)−1

2

π(y) =kµ/π−1k22,π. Defn. Rel. Entropy:

D(Pn(x,·)kπ) =P

yPn(x, y) logPn(x, y)/π(y)= Entπ(knx), where Entπ(f) = EπflogfEπflog(Eπf).

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Various Mixing Times

A Key Question. Howquicklydoes it converge to 1?

The total variation, relative entropy andL2 mixing times are defined as follows, (using worst-case starting point).

τ() = min{n:∀x∈Ω, kPn(x,·)−πktv≤} τD() = min{n:∀x∈Ω, D(Pn(x,·)kπ)≤} τ2() = min{n:∀x∈Ω, kkn(x,·)−1k2,π≤}

I

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List of Techniques

Functional Analytic,IsoperimetricApproaches:

I Spectral Gap,Conductance (Cheeger-type constants)

I Entropy Constant (Dai Pra-Paganoni-Posta, Gao-Quastel, Bobkov-T.), Log-Sobolev Constant (Gross, Bakry-Emery, Diaconis-Saloff-Coste, Miclo, ...)

I Spectral Profile (Goel-Montenegro-T.) and Isoperimetric Profile(Kannan-Lovasz) ProbabilisticApproaches:

I Coupling, Path coupling (Bubley-Dyer),

Coupling From the Past (Propp-Wilson), Evolving Sets (Morris-Peres)

Other Techniques

I Decomposition and Comparison

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Treatment with Dirichlet Forms

The following will be done on the white board:

I Motivating definitions of spectral gap, entropy constant, ...

I Comparison of Dirichlet Forms

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An Amazing Conjecture

[Due to Aldous-Diaconis ’92].

Conjecture. For everyG,λIP(G) =λRW(G).

IP: Start withnlabeled particles on ann-vertex graph. For each edgeij independently and at rate 1, interchangethe particlesiand j. (This is reversible w.r.t. uniform distribution onn! configurations. of particles.)

RW: Observe a single particle above, thus obtaining a (continuous) time simple random walk onG.

I Known to hold for G=Cn, Kn, Trees, and a bit more ...

I Asymptotically forG:d-dimensional grid of side length n.

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The logarithmic factor: some examples

I Entropy Constant vs Log-Sobolev constant:

random transpositions, k-sets of an n-set

I The Pollard Rho walk

I The Thorp shuffle

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Coupling

Colorings on graphs : a case study

Refer to the file coloring.pdf, which covers Jerrum’s coupling and a bit more.

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Next Topic

Temporal vs SpatialMixing

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Temporal Mixing and Uniqueness of Gibbs

Typically one expects fast mixing of Glauber (and other local) dynamics in the uniqueness regime, and (exponentially) slow mixing in the non-uniqueness regime:

Examples include: OnZd,

Ising at low-temperature (largeβ >0), Hard-core lattice gas (i.e., weighted independent sets), etc.

However, Glauber dynamics onZdin the multiphase region is still poorly understood: F. Martinelli’s problem with the +boundary condition onZ2.

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An Amazing Theorem

[Martinelli-Olivieri-Schonman’94].

For the Ising model on a finite box of sizeninZ2, the Glauber dynamics satisfies:

I If T > Tc, then mixing time =O(nlogn)

I If T < Tc, then mixing time = exp(Ω(√ n)).

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A notion of Strong Spatial Mixing

Defn. A system hasSSMif∃ constantsβ and α >0, such that for any two subsets Λ,Ψ, with Λ⊂Ψ, any site u∈∂Ψ, and any pair of bdry configs. τ andτu differing only atu,

d µτΨ, µτΨu

|Λ≤β|Λ|exp(−α dist(u,Λ)).

I Weak Spatial Mixing: replaced(u,Λ) by d(∂Ψ,Λ); holds up to (and corresponds to) uniqueness of Gibbs etc.

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Optimal Temporal Mixing

[In words: Glauber dynamics on a finite volume Ψwith ANY bdry config. reaches stationarity in timeO(nlogn), n=|Ψ|.]

Defn. Glauber has OTM if∃ const. b, c >0 such that

∀Ψ⊂⊂ Zd with any bdry config. for any two instances (Xt) and (Yt) of the chain, anyk∈ Z+,

d(Xkn, Ykn)≤bn e−ck.

I e.g., if entropy constantρ0≥c0/n, then OTM holds.

Thm. [Stroock-Zegarlinski’92]. For spin systems onZd, with nearest neighbor interactinos, OTM holds iff SSM holds.

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Path of Disagreement in Bdd-degree Graphs

Key Lemma[Zegarlinski’ ??, van den Berg’93] (DSVW’02):

G= (V, E), |V|=n, max degree ∆.

LetA, B⊂V arbitrary with r= dist(A, B).

Xt, Yt : two copies of Glauber on GwithX0≡Y0 off ofA.

Then for pos. integerk≤r/[(∆−1)e2], if we run for T =kn steps, we have

P(XT(B)6=Yt(B))≤4 min{|A|,|B|}(∆−1)ek r

r

. The probability is under “identity coupling” of (Xt),(Yt).

I Cor. IfT =kn,r= dist(A, B)≥(∆−1)e2k, P(XT(b)6=YT(B))≤4|A|e−dist(A,B). WMA assumed above that|A| ≤ |B|.

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OTM implies SSM in Z

d

Run chain (Xt) with the stat. measureµτΨ, and chain (Yt) with the stat. measureµτΨu, starting withX0=Y0 (in the same configuration of spins).

Stop at timet= (2d−1)edist(u,Λ)2 n(say). Then

I

d(µτΨ, µτΨu)≤d(µτΨ, Xt) +d(Xt, Yt) +d(µτΨu, Yt).

I

d(Xt, Yt)|Λ≤4|Λ|e−dist(u,Λ), sincedisagreement has not percolated!

I OTOH,

d(µτΨ, Xt)|Λ≤b0|Λ|exp

− c0

(2d−1)e2 dist(u,Λ) , due to OTM.

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OTM implies SSM in Z

d

Run chain (Xt) with the stat. measureµτΨ, and chain (Yt) with the stat. measureµτΨu, starting withX0=Y0 (in the same configuration of spins).

Stop at timet= (2d−1)edist(u,Λ)2 n(say). Then

I

d(µτΨ, µτΨu)≤d(µτΨ, Xt) +d(Xt, Yt) +d(µτΨu, Yt).

I

d(Xt, Yt)|Λ≤4|Λ|e−dist(u,Λ), sincedisagreement has not percolated!

I OTOH,

d(µτΨ, Xt)|Λ≤b0|Λ|exp

− c0

(2d−1)e2 dist(u,Λ) , due to OTM.

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OTM implies SSM in Z

d

Run chain (Xt) with the stat. measureµτΨ, and chain (Yt) with the stat. measureµτΨu, starting withX0=Y0 (in the same configuration of spins).

Stop at timet= (2d−1)edist(u,Λ)2 n(say). Then

I

d(µτΨ, µτΨu)≤d(µτΨ, Xt) +d(Xt, Yt) +d(µτΨu, Yt).

I

d(Xt, Yt)|Λ≤4|Λ|e−dist(u,Λ), sincedisagreement has not percolated!

I OTOH,

d(µτΨ, Xt)|Λ≤b0|Λ|exp

− c0

(2d−1)e2 dist(u,Λ)

, due to OTM.

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Proof Sketch of Key Lemma

Observe: if sitev at timeT has different spins, then there must exist a path of disagreement fromA tov: i.e., ∃ path

v0, v1, . . . , vl=v and 0< t1 < t2· · ·< tl≤T, s.t. v0 ∈A, and vi updated at timeti.

I Pr[v0, . . . , vl]≤ Tl

(1/n)l,for a fixed path.

I No. of paths of lengthl≥r fromA toB is at most min{|A|,|B|}∆(∆−1)l−1.

I So Pr(Xkn(B)6=Ykn(B))≤ |A|

(∆−1)

P

l≥r

(∆−1)ek

l

l

≤4|A|(∆−1)ek

r

r

,

≤4|A|e−r, ifr ≥(∆−1)e2k.

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But ... what about the sub-exponential growth?

Have we used the fact that we were inZd?

I In the OTM bound, we really should have used a bound of d(µτΨ, Xt)|Λ≤b0|Ψ|exp

− c0

(2d−1)e2 dist(u,Λ) , rather than the smaller bound with |Λ|in place of|Ψ|, since the chain was run on Ψ rather than on Λ.

I So to justify the “fudge” we actually need an extra lemma:

this would argue that we could first restrict our attention to dynamics inside a volume Λk which contains all sites within distance ˆk= (2d−1)e2k from some site in Λ.

I The choice of ˆk still lets us use the path of diagreement argument on not reachingΛ.

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Fudge contd...

I The above would then give a bound of |Λk|e−c

0dist(u,Λ) (2d−1)e2

(plus another exponential term such as, e−ck|Λk|, for the probability of missing Λk, while choosing sites at random from Ψ: using Chernoff’s bound, e.g.)

I Now crucially, the “sub-exponential growth” of Zd can be used in saying |Λk| ≤

2[2d−1]e2kd

|Λ|and absorbing it into :

k|e−c0···≤ |Λ|e−c

00 dist(u,Λ) (2d−1)e2 .

Details in [Dyer-Sinclair-Vigoda-Weitz’02].

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Spectral Gap and Correlations on a general graph

At∞ temperature, where distinct vertices are independent, the Glauber dynamics on a graph ofnvertices reduces to an

(accelerated by a factor ofn) random walk on a discrete n-cube where the relaxation is Θ(1).

The next result shows that at any temperature where such fast relaxation takes place, a fairly strong form of independence holds. The setting is that of any graph ofbounded degree.

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Spectral Gap and Correlations – 2

Thm. [Ber-Ken-Mos-Per’05] LetGbe bdd. degree graph, and letσr be config. on the setGr of all vertices at distance r from a fixed vertexv. Let therelaxation timeof Glauber dynamics onGr satisfyτ2(Gr) =O(1). Then the Gibbs distribution on Gr has the following property:

For any fixed setA of vertices,∃cA>0, s.t. for r large enough, Cov[f, g]≤e−cArp

Var(f)Var(g),

providedf(σ) depends only onσA andg(σ) depends only onσr. (Equivalently,∃c0A>0 s.t. the mut. inform. I(σA, σr)≤e−c0Ar.)

I Proof uses disagreement percolation and coupling as before.

I Thm could hold also when there are multiple Gibbs measures– e.g., the Ising model in the “intermediate regime” – for 1−2≤1/√

b, one gets limr→∞I[σ0, σr] = 0.

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An Open Question

Question[BKMP’05]:

For the Ising model (with free boundary conditions and no external field) on a general graph of bounded degree, does the converse of the above theoremhold?

That is, does uniform exponential decay of point-to-set correlations imply a uniform spectral gap?

I F.M. : Not always true – fails in certain lattices if plus bdry conditions are allowed.

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Broadcast Process on a Tree

Model

I T infinite rooted tree where each node has ∆ children.

I k colors,σv ∈[k] is the color at vertexv.

I Choose σr at the root uniformly at random.

I Broadcast process.

I Independent Markov chain on each edge (u, v). P(i, j) = P[σu=j|σv=i]

I Proper colorings.

P(i, j) = δi6=j k1

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Broadcast Process on a Tree

Model

I T infinite rooted tree where each node has ∆ children.

I k colors,σv ∈[k] is the color at vertexv.

I Choose σr at the root uniformly at random.

I Broadcast process.

I Independent Markov chain on each edge (u, v).

P(i, j) = P[σu=j|σv=i]

I Proper colorings.

P(i, j) = δi6=j k1

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Broadcast Process

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Broadcast Process

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Broadcast Process

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Broadcast Process

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Broadcast Process

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Broadcast Process

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Broadcast Process

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Reconstruction

How doesµ(·|X) behave as a function of a typical coloringX?

∃c, lim

`→∞EX(|µ(c|X)− 1 k|)>0

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Applications I -Phylogeny

I [Mossel ’04], [Daskalakis-Mossel-Roch ’06 ] Evolution of DNA, phylogenetic reconstruction.

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Applications II - Convergence of Markov Chains

I [Kenyon-Mossel-Peres ’01], [Berger-K-M-P ’05]

O(n) relaxation time for Glauber dynamics on bounded degree graphs implies non-reconstruction.

I [Martinelli-Sinclair-Weitz ’03] Exponential decay of correlation on the tree impliesO(nlnn) convergence for Glauber dynamics when “uniqueness” holds.

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Applications III - Replica Symmetry Breaking Transition

I [Achlioptas, Coja-Oghlan ’08]

(1/2 +o(1))∆/ln ∆< kd<(1−o(1))∆/ln ∆.

I Conjecture that kd is the same as the reconstruction threshold for trees. There is very recent work in this direction by [Montanari-Restrepo-T.]

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Bounds on the Threshold for Non-reconstruction

I [Brightwell-Winkler’00]

Non-Reconstruction fork≥∆ + 1

I [Jonasson’02]

fork≥∆ + 1: root is unbiased for any fixed coloringof the leaves; (thus the Gibbs measure is unique.)

I [Mossel-Peres ’03]

Reconstruction when k≤ (1−ε)∆ln ∆ .

I [B-Vera-Vigoda-Weitz ’08]

Non-reconstruction for k > (1+ε)∆ln ∆ for ∆≥∆ε.

I [Sly ’08]

Non-reconstruction for ∆≤k(lnk+ ln lnk+ 1 +o(1)).

Reconstruction for ∆≥k(lnk+ ln lnk+ 1−ln 2−o(1)).

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Bounds on Mixing

I [Bhatnagar-Vera-Vigoda-Weitz ’08]

O(nlogn) mixing for a certain Block dynamics, by bounding the entropy constant of the dynamics for k > (1+ε)∆ln ∆ for ∆≥∆ε.

I [Goldberg-Jerrum-Karpinsky ’08]

Glauber: For fixed kand ∆ so that 3≤k≤∆/2 log(∆):

Upper bound ofnO(∆/log ∆) and a lower bound of nΩ(∆/klog ∆).

Problem 1. More precise bounds on the performance of Glauber “around”k= ∆/log ∆ ,especially in the range

∆/2 log ∆< k <2∆/log ∆ . Problem 2. SSM for k≥∆ + 1.

42/42

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