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Spin systems with hyperbolic symmetry a survey · 2022-prob-192

Special processes · ICM 2022: Spin systems with hyperbolic symmetry a survey

MSC 60 · Probability theory and stochastic processes

Inferred from the community field; finer classification has not been reviewed.

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Problem statement

For an integer d≥3d\ge 3, let ΛN=(Z/LNZ)d\Lambda_N=(\mathbb Z/L^N\mathbb Z)^d be a sequence of discrete tori with their nearest-neighbor edge sets, where L≥2L\ge 2. For β>0\beta>0, define the arboreal-gas probability measure on spanning forests FF of ΛN\Lambda_N by

PβΛN(F)=1ZN,β β∣E(F)∣, \mathbb P^{\Lambda_N}_\beta(F)=\frac{1}{Z_{N,\beta}}\,\beta^{|E(F)|},

where E(F)E(F) is the edge set of FF and ZN,βZ_{N,\beta} is the normalizing constant. Let Pβ\mathbb P_\beta be an infinite-volume local weak limit of these measures on spanning forests of Zd\mathbb Z^d. Prove or disprove that, for every d≥3d\ge 3 and all sufficiently large β\beta, a forest sampled from Pβ\mathbb P_\beta almost surely has exactly one infinite connected component when d∈{3,4}d\in\{3,4\} and infinitely many infinite connected components when d>4d>4.

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Question 1

For an integer d≥5d\ge 5, let ΛN=(Z/LNZ)d\Lambda_N=(\mathbb Z/L^N\mathbb Z)^d be discrete nearest-neighbor tori, where L≥2L\ge2, and let the arboreal-gas measure with parameter β>0\beta>0 assign each spanning forest FF of ΛN\Lambda_N probability proportional to β∣E(F)∣\beta^{|E(F)|}. Prove that there exists β0(d)<∞\beta_0(d)<\infty such that, for every β>β0(d)\beta>\beta_0(d), every infinite-volume local weak subsequential limit of these measures is almost surely supported on forests of Zd\mathbb Z^d having infinitely many infinite connected components.

Source reference: Noah Halberstam and Tom Hutchcroft, “Uniqueness of the infinite tree in low-dimensional random forests,” Probability and Mathematical Physics 5(4) (2024), 1185–1216, unnumbered conjecture in the abstract and introduction, DOI: 10.2140/pmp.2024.5.1185, arXiv:2302.12224.

Progress & source assessment

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The problem is unresolved. The assertion is proved for d=3,4d=3,4, but the claim that there are infinitely many infinite trees for d≥5d\ge 5 remains open.

Roland Bauerschmidt, Nicholas Crawford, and Tyler Helmuth, “Percolation transition for random forests in d⩾3d\geqslant 3,” Inventiones Mathematicae 237 (2024), 445–540, doi:10.1007/s00222-024-01263-3, arXiv:2107.01878. Their low-temperature percolation theorem proves, for every d≥3d\ge3 and all sufficiently large β\beta, the existence of an infinite tree in every subsequential local weak limit of the arboreal gas on tori (and gives quantitative long-range correlation asymptotics). It does not determine the number of infinite trees.

Noah Halberstam and Tom Hutchcroft, “Uniqueness of the infinite tree in low-dimensional random forests,” Probability and Mathematical Physics 5 (2024), 1185–1216, doi:10.2140/pmp.2024.5.1185, arXiv:2302.12224. Theorem 1.1 proves that every translation-invariant arboreal-gas Gibbs measure on Zd\mathbb Z^d has at most one infinite tree when d=3,4d=3,4. Combining this with the preceding percolation theorem, Corollary 1.2 proves that every subsequential weak limit of the low-temperature arboreal gas on dd-dimensional tori has exactly one infinite tree for d=3,4d=3,4. The same paper proves a resampling description and one-endedness of all infinite trees in every dimension, but explicitly leaves the prediction of infinitely many infinite trees in d≥5d\ge5 as a conjecture. No later proof or counterexample resolving that remaining high-dimensional case was found.

References
  1. [1]ICM Conjectures · 2022-prob-192 link
  2. [2]ICM 2022: Spin systems with hyperbolic symmetry a survey link

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