# P2700: Exact heat-bath spectral gap on the six by six Ising torus

- ID: `P2700`
- Reference: `ising-six-torus-heat-bath-gap`
- Page: https://theoremdb.org/statements/P2700
- Record maturity: Reviewed problem with recorded work

## Problem

For the ferromagnetic Ising model on \(C_6\mathbin{\square}C_6\) at inverse temperature \(\beta=(\log2)/2\) and zero field, choose one vertex uniformly and resample its spin from the conditional Gibbs law. Determine the exact spectral gap of this \(2^{36}\)-state Markov chain.

### Remarks

- **Remark.** The Hamiltonian is minus the sum of sigma_u sigma_v over the 72 torus edges.
- **Remark.** The spectral gap is 1-lambda_2, where lambda_2 is the second-largest eigenvalue of the reversible transition operator.

### What counts as a solution

- Give the gap as an exact algebraic value or a rational isolating interval, with a symmetry-block characteristic or verified eigenvalue certificate covering every sector.

## Status

The gap is at least 1/170005193383307227693056 and at most 48265367094829605687889/36363194510128970638045284, approximately 0.00132731372325897517. [1](#reference-1)

## Work

### Evidence for the current status

**Computation 1 (A certified rational interval for the six-torus heat-bath gap).** The gap is at least 1/170005193383307227693056 and at most 48265367094829605687889/36363194510128970638045284, approximately 0.00132731372325897517.

Write \(\gamma\) for the discrete-time spectral gap under the convention in the problem. The certified interval is
\[
\frac{1}{170005193383307227693056}
\leq \gamma \leq
\frac{48265367094829605687889}{36363194510128970638045284}
=0.00132731372325897517\ldots.
\]
The lower endpoint equals \(1/(36\,2^{72})\). The upper endpoint is an exact Rayleigh quotient for total magnetization \(M=\sum_v\sigma_v\). This trial function lies in the spin-flip-odd, translation-invariant symmetry sector. Its projected transition eigenvalue is
\[
\frac{36314929143034141032357395}{36363194510128970638045284}.
\]

For the upper bound, reversibility and the heat-bath projection identity give
\[
\langle M,(I-P)M\rangle_\pi
=\mathbb E_\pi\!\left[\operatorname{Var}(\sigma_0\mid\sigma_{V\setminus\{0\}})\right].
\]
The exact transfer calculation finds
\[
\sum_\sigma 2^{a(\sigma)}M(\sigma)^2
=273793464546853425980576256,
\]
where \(a(\sigma)\) is the number of agreeing edges. With common conditional-variance scale \(7225=25\cdot17^2\), it also finds
\[
\sum_\sigma 2^{a(\sigma)}\,7225\operatorname{Var}(\sigma_0\mid\sigma_{V\setminus\{0\}})
=2625635969958730549421161600.
\]
Their quotient gives the stated upper endpoint.

For the lower bound, compare with independent random-scan heat bath under the uniform measure. If \(m\) and \(M_0\) are the minimum and maximum density ratios of the Ising measure against uniform measure, edge conductances give \(\mathcal E_P\geq m\mathcal E_0\), while \(\operatorname{Var}_\pi f\leq M_0\operatorname{Var}_0f\). The independent chain has gap \(1/36\). The Gibbs weights are proportional to \(2^{a(\sigma)}\), and the bipartite torus realizes both \(a=0\) and \(a=72\). Hence \(m/M_0=2^{-72}\), proving the lower endpoint.

The calculation supplies rigorous bounds. Determining the exact second eigenvalue still requires a certificate that covers every symmetry sector.

### Background and intake notes

The raw matrix has 68719476736 states. Exact symmetry reduction and transfer methods are needed beyond direct storage.

- Original intake status: Novelty remains unverified. Finite-volume Ising gaps are classical, and no primary-source status audit was completed for this exact size and rational-temperature choice.
- At this beta, the five possible conditional probabilities are 1/17,1/5,1/2,4/5,16/17, so the transition matrix is rational and exact certificates are possible.
- Spin-flip, translation, and dihedral symmetries should block-diagonalize the operator before rational reconstruction.
- A power-iteration decimal is evidence for one eigenmode. It does not exclude a larger eigenvalue in another symmetry sector.

- Recorded example: The same convention on the 3 by 3 and 4 by 4 tori gives approximate gaps 0.0081675847 and 0.0035257034.

### Prior approaches

- **Route 1** (supported): The checked sources cover spectral-gap comparison and high-temperature torus dynamics. The audit found no published characteristic polynomial or exact gap for this parameter choice. [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (reproduced): A standard-library Python program contracts the torus exactly, computes the one-dimensional projected eigenvalue, and checks fixed matrix hashes. [1](#reference-1)

### Computational notes

- Implicit reversible power iteration over all states reproduced gaps 0.008167584663204663 for side 3 and 0.003525703353460252 for side 4. Direct enumeration also verified that every heat-bath probability is one of the five stated rationals.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `ising-six-torus-heat-bath-gap`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Eyal Lubetzky and Allan Sly, “Cutoff for the Ising model on the lattice”. Inventiones mathematicae 191(3) (2013), 719-755. DOI 10.1007/s00222-012-0404-5. Richard Holley and Daniel Stroock, Logarithmic Sobolev Inequalities and Stochastic Ising Models, Journal of Statistical Physics 46 (1987), 1159-1194, DOI 10.1007/BF01011161; Fabio Martinelli, Lectures on Glauber Dynamics for Discrete Spin Models, Lecture Notes in Mathematics 1717 (1999), 93-191, DOI 10.1007/978-3-540-48115-7_2; Eyal Lubetzky and Allan Sly, Cutoff for the Ising Model on the Lattice, Inventiones Mathematicae 191 (2013), 719-755, DOI 10.1007/s00222-012-0404-5 https://doi.org/10.1007/s00222-012-0404-5
   - Also cited at Exact transfer certificate in ising6-artifact-row-transfer-rayleigh and comparison proof in this record
   - Also cited at Inline Python 3 exact computation executed on 2026-07-25
   - journal_article; primary source; version of record; checked 2026-08-01
   - Source use: original_summary
   - For Exact heat-bath spectral gap on the six by six Ising torus: The checked sources cover spectral-gap comparison and high-temperature torus dynamics. The audit found no published characteristic polynomial or exact gap for this parameter choice.
   - Source named by the research packet.