OpenFrontiers
Join
← Back to the index
OP-8753Status unassessedProbability

Spin glass with i.i.d. couplings

Markov chain mixing times · Historical workshop question

MSC 60 · Probability theory and stochastic processes

Editorial mapping of AIMPL source category: Probability. http://aimpl.org/markovmixing/ Not an author-supplied MSC code.

Historical list · status unassessedAs of 2026-09-08Source & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Consider the spin glass model on Znd\mathbb{Z}_n^d with i.i.d. couplings, i.e. with Hamiltonian given by

Hn(σ)=∑(i,j)∈EnJijσiσj ,H_n(\sigma)= \sum_{(i,j)\in E_n} J_{ij}\sigma_i\sigma_j\, ,

where J=(Jij)J=(J_{ij}) is a random symmetric matrix with i.i.d. ±1\pm 1 entries. The Gibbs distribution is given by

μn(σ)=1Znexp⁡(βHn) .\mu_n(\sigma)= \frac{1}{Z_n}\exp(\beta H_n)\, .

Now consider the Glauber dynamics. Does there exist β0\beta_0 and ε>0\varepsilon>0 such that, conditionally on JJ, for all β≥β0\beta\geq\beta_0, the mixing time of this chain is at least exp⁡(nε)\exp(n^\varepsilon)?

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

Workshop introduction

In general, we will use the following notation in the problem list. We consider aperiodic irreducible Markov chains on a state space, Ω\Omega with unique invariant (stationary) distribution π\pi. In the discrete time setting we allow the matrix (P)x,y∈Ω(P)_{x,y\in\Omega} with Pxy=P(x,y)P_{xy}=P(x,y) to denote the transition matrix of the Markov chain.

A chain is reversible if π(x)P(x,y)=π(y)P(y,x)\pi(x)P(x,y)=\pi(y)P(y,x) for all x,y∈Ωx,y\in\Omega. Q(x,y)=π(x)P(x,y)Q(x,y)=\pi(x)P(x,y) is the edge measure on (x,y)(x,y). We denote by t\mboxmixt_{\mbox{mix}} the total variation mixing time defined as

t\mboxmix(ϵ)=inf⁡{t:sup⁡ω0∈Ω∥Pt(ω0,⋅)−π(⋅)∥TV≤ϵ} ,t_{\mbox{mix}}(\epsilon)=\inf\{t:\sup_{\omega_0\in\Omega}\|P^t(\omega_0,\cdot)-\pi(\cdot)\|_{TV}\leq \epsilon\}\,,

and set t\mboxmix=t\mboxmix(ϵ)t_{\mbox{mix}}=t_{\mbox{mix}}(\epsilon), where the total variation distance

∥μ−ν∥TV=12∥μ−ν∥ℓ1 .\|\mu-\nu\|_{TV}=\tfrac 12 \|\mu-\nu\|_{\ell^1}\,.

Progress & source assessment

Historical workshop record. Current open/solved status has not been independently reassessed.

The source supplies no status note for this question.

References
  1. [1]Markov chain mixing times link

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.