Glauber Dynamics on Trees and Hyperbolic Graphs Elchanan Mossel, Microsoft Research joint work with Claire Kenyon, L.R.I. Paris IX Yuval Peres, U.C. Berkeley http://research.microsoft.com/~mossel/ 11/3/2018

Glauber dynamics for coloring G = (V,E) a finite graph of n vertices, where all degrees ≤ D. Want to sample coloring with q > D colors. Algorithm to sample “proper” coloring σ of G Start with a proper coloring σ. Repeat the following: Pick a vertex v uniformly at random, and update the color σ(v) to be uniformly chosen from the set [q] \ {σ(w): v ~ w}. Converge to uniform coloring; but how fast? (Vigoda 2000). Does speed depend on the Geometry of G? 11/3/2018

The Ising Model It is easier to analyze the Ising model: Let G=(V,E) be a finite graph with n vertices. The Ising model on G is a probability measure (Gibbs distribution) on the space of configurations σ from V to {-1,1} such that T = 1/β is the “inverse temperature”, and Z is the “partition function”. Both models have “local” constraint. 11/3/2018

Glauber dynamics for Ising models Algorithm to sample from the Gibbs dist. Start with a configuration σ. Repeat the following: Pick a vertex v uniformly at random, and update σ(v) according to the conditional probability given {σ(w): w ~ v}. Converge to Gibbs distribution; but how fast? Does speed depend on the Geometry of G? Mostly studied when G is a box in Zd (Martinelli lecture notes). 11/3/2018

Mixing and relaxation times Glauber dynamics defines a d.s. matrix with spectrum 1 > γ1 > … >. The “spectral gap” of the dynamics is 1-γ1 The “relaxation time” is τ2 = 1/(1 - γ1). The “total-variation” distance between μ and ν is: Let P(t,σ) be the distribution of the dynamics started at σ at time t. The “mixing time” of the dynamics is defined by: In general: 11/3/2018

General picture When β is small (large number of colors), τ2 = Θ(n), and τ1 = Θ(n log n). When β is large (small number of colors), mixing time may be large. In [-L,L] d, when β < βc, the mixing time is exp(Θ(Ld-1 )) (physics literature) n = Ld 11/3/2018

Simple graphs The Ising model on the line graph has mixing time n log n for all β. There exists β(D,α) such that if G=(V,E) is an α-expander, then for all β > β(D,α), the mixing time of the Ising model on G is exp(Θ(n)). n = |V| 11/3/2018

Bounding relaxation by exposure Following the “canonical path” method of Jerrum-Sinclair (& Martinelli), We define the exposure, ex(G) of a graph G=(V,E) of maximal degree Δ, as the smallest integer for which there exists a labeling v1,…,vn of V s.t. for all 1 < k < n, the number of edges connecting {v1,…,vk} to {vk+1,…,vn} is at most ex(G). THM[KMP]: For Ising-Glauber dynamics on G: For Coloring-Glauber dynamics on G, when q > Δ + 1: 11/3/2018

Application to Ising model in Zd Labeling order In Z1, gives τ2 = O(L2) at all β (truth is O(L)). In Zd, d > 1, gives τ2 = (exp(O(Ld-1))), which is correct (up to constant factor in the exp) when β is large. Open problem: Find properties of graphs which imply similar lower bounds. 11/3/2018

Trees and hyperbolic graphs For the binary tree T, using DFS order, ex(T) is the height of T, and therefore the relaxation (mixing) time is poly(|T|) for all β. Similarly, we prove polynomial mixing time for balls in graphs of hyperbolic tilings. 11/3/2018

Remarks For trees, it easy to generate the Gibbs distribution rapidly, in a top-bottom manner. For hyperbolic graphs, our results give a polynomial time algorithm, for sampling colorings when q > Δ + 1; and Ising models for all β. Folklore belief: In the “ordered phase” (1 Gibbs measure) – τ2 = poly(n) , in the “unordered phase” (multiple Gibbs measure) – τ2 = super-poly(n). For trees and hyperbolic tilings, when β is large, we have ∞ Gibbs measures but τ2 = poly(n) ??? 11/3/2018

The Ising model on the binary tree The (Free)-Ising-Gibbs measure on the tree T: Set σr, the root spin, to be +/- with probability ½. For all pairs of (parent, child) = (v, w), set σw = σv, with probability 1 – ε, independently for all pairs (v,w). + + + + - + + - + - + - + + + 11/3/2018

Relaxation time for the binary tree In KMP we prove that: mutual information: H(σ∂) + H(σr)) - H(σr,σ∂) Temp 1 - 2ε σr | σ∂≡ 1 Uniq I(σr,σ∂) Free measure τ2 high < 1/2 unbiased V → 0 extremal O(n) med. (1/2,1/√2) biased X low > 1/√2 Inf > 0 Non-ext n1+Ω(1) freeze 1 – o(1) nΘ(β) Uniqueness phase transition plays no role for relaxation. Extremality phase transition linear / non-linear relaxation. 11/3/2018

Temporal mixing spatial mixing Thm [KMP]: Let G be an ∞-graph of bounded degree; (Gr) balls of radius r around o. Consider nearest-neighbor particle system (e.g. Ising; Coloring) on G s.t. Glauber dynamics on Gr satisfy τ2 = O(|Gr|). Then, for any finite sets A, I((σv)v in A , (σv)|v| > r) = exp(-Ω(r)). Equivalently, if f is a function of (σv)v in A and g a function of (σv)|v| > r, then Cov(f,g) < exp(-Ω(r))Var(f) Var(g). Open problem: spatial temporal? g r A f 11/3/2018

If we find a way to replace Qt by Pt, for t > c*r, we are done. Proof sketch We bound E[f*g] when E[f] = E[g] = 0. Consider two dynamics on Gr: Glauber dynamics where moves are conditioned on the boundary. Let Qt[f](σ) = E[f(σt)], where σt is σ after t updates of this dynamics. Glauber dynamics where moves are independent of the boundary. Let Pt[f](σ) = E[f(σt)], for this dynamics. Since g is independent of the configuration in Gr, E[f*g] = E[Qt[f]*g] ≤ |Qt[f]|2 |g|2. We know that: |Pt[f]|2 ≤ τ2-t|f|2. If we find a way to replace Qt by Pt, for t > c*r, we are done. 11/3/2018

Paths of disagreement It remains to estimate |Ptf – Qtf|2. Note that Pt[f] = Qt[f], unless there exists a path v1,…vk, with |v1| > r and vk in A, s.t. vi is updated after vi-1,and vk is updated before time t. Since all updates are contractions in L2: |Ptf – Qtf| ≤ |f| P[ ] When t = c*r, for small c > 0, = |f| exp(-Ω(r)). (similar to v.d.Berg proof) g r A f 2 2 2 11/3/2018

The ternary tree in low temperatures The exposure result, or a recursive argument prove that τ2 = poly(n), for all n and β. To obtain lower bounds on τ2, we find bottlenecks in the state space (easy part of Cheeger/conductance estimates): for 11/3/2018

The ternary tree in low temperatures In order to obtain τ2 > n1+Ω(1) in low temperatures A = {σ : majority of spins in level n of σ are +}. In order to obtain τ2 > nΘ(β) for freezing temp: A = {σ : recursive maj of spins in level n of σ are +}. 11/3/2018

The tree in med. & high temperatures The analysis uses “block dynamics”: we update sub-trees of up to h = h(β) levels, which include all sub-trees of h levels, and all sub-trees of ≤ h levels which contain leaves, or the root. The block dynamics and the single-site dynamics have up to a constant (which depends on h) the same τ2 (This is well known, e.g. Martinelli. Not known if the same holds for the mixing time) h 11/3/2018

The tree in med. & high temperatures We define a weighted hamming metric for the b-ary tree: , where |v| = distance from v to the root. Let σ’ be σ after an update. It suffices to construct a coupling s.t. E[d(σ’, η’)] ≤ (1 – c/n) d(σ, η). This implies by a general principle (Chen), that τ2 = Ω(n). By the method of “path coupling” (Jerrum-Sinclair), suffices to show the contraction when σ and τ differ in one spin. σ d(σ,τ) τ In the top line all vertices differ in one spin only σ’ τ’ 11/3/2018

The tree in med. & high temperatures Let σ and η differ at a single site v. There are 4 cases to consider, depending on the relative location of the updated block and v: It turns out that for the Ising model, in the last two cases Ed(σ’,η’), may be bounded by Ed(σ’,η’) for and (with no other boundary conditions). v v v v 11/3/2018

Summary We show how the exposure of a graph gives an upper bound on the relaxation time for Glauber dynamics for Ising models and colorings of the graph. For trees and hyperbolic graphs, the relaxation time is always polynomial in the size of the graph. For the tree, the uniqueness phase-transition plays no role for the relaxation time, and the extremality phase transition corresponds to linearity of τ2 in n. Linearity of τ2 in n always implies extremality. 11/3/2018