Scaling limits of the three-dimensional uniform spanning tree and associated random walk

Angel, Omer; Croydon, David A.; Hernández-Torres, Saraí; Shiraishi, Daisuke

doi:10.1214/21-aop1523

Cited by 11 publications

(4 citation statements)

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“…Beyond the questions above, it would be interesting to analyse more detailed geometric aspects of the arboreal gas. For example, can one construct scaling limits as has been done for some spanning tree models [3,5,6,48]?…”

Section: Infinite Volume Behaviour and Relation To The Uniform Spanni...mentioning

confidence: 99%

Percolation transition for random forests in $d\geqslant 3$

Bauerschmidt,

Crawford,

Helmuth

2024

Invent. math.

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The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor $\beta >0$ β > 0 per edge. It arises as the $q\to 0$ q → 0 limit of the $q$ q -state random cluster model with $p=\beta q$ p = β q . We prove that in dimensions $d\geqslant 3$ d ⩾ 3 the arboreal gas undergoes a percolation phase transition. This contrasts with the case of $d=2$ d = 2 where no percolation transition occurs.The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane $\mathbb{H}^{0|2}$ H 0 | 2 . This latter model can be thought of as the 0-state Potts model, with the arboreal gas being its random cluster representation. Unlike the standard Potts models, the $\mathbb{H}^{0|2}$ H 0 | 2 model has continuous symmetries. By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures. In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit. Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori.

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Section: Infinite Volume Behaviour and Relation To The Uniform Spanni...mentioning

confidence: 99%

Percolation transition for random forests in $d\geqslant 3$

Bauerschmidt,

Crawford,

Helmuth

2024

Invent. math.

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“…This was later extended to full convergence in a result of Holden and Sun [ 18 ]. On , much less is known, however the breakthrough works of Kozma [ 21 ] and Li and Shiraishi [ 28 ] on subsequential scaling limits of LERW enabled Angel, Croydon, Hernandez-Torres and Shiraishi [ 4 ] to show GHP convergence of the rescaled UST along a dyadic subsequence. Their scaling factors are given in terms of the LERW growth exponent in three dimensions, which was shown to exist by Shiraishi [ 37 ].…”

Section: Introductionmentioning

confidence: 99%

The GHP Scaling Limit of Uniform Spanning Trees in High Dimensions

Archer,

Nachmias,

Shalev

2024

Commun. Math. Phys.

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We show that the Brownian continuum random tree is the Gromov–Hausdorff–Prohorov scaling limit of the uniform spanning tree on high-dimensional graphs including the d-dimensional torus $${\mathbb {Z}}_n^d$$ Z n d with $$d>4$$ d > 4 , the hypercube $$\{0,1\}^n$$ { 0 , 1 } n , and transitive expander graphs. Several corollaries for associated quantities are then deduced: convergence in distribution of the rescaled diameter, height and simple random walk on these uniform spanning trees to their continuum analogues on the continuum random tree.

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“…Starting from various fine properties of LERW and Kozma's scaling limit result ( [21], [19], [22], [16]), the authors showed ( [17]) that LERW converges weakly as a process with time parametrization, which greatly strengthens Kozma's result. This improvement plays a key role in [1] to prove that the three-dimensional uniform spanning tree (UST) converges weakly as a metric space endowed with the graph distance.…”

Section: Introductionmentioning

confidence: 99%