Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process

Bauerschmidt, Roland; Helmuth, Tyler; Swan, Andrew

doi:10.1214/19-aop1343

Cited by 14 publications

(20 citation statements)

References 33 publications

(71 reference statements)

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“…We also use this fermionic representation, but our results rely in an essential way on the new observation that this model is most naturally connected to a sigma model taking values in a hyperbolic superspace. Similar sigma models have recently received a great deal of attention due to their relationship with random band matrices and reinforced random walks [6,21,44,45]. We will discuss the connection between our techniques and these papers after introducing the sigma models relevant to the present paper.…”

Section: The Arboreal Gas and Uniform Forest Modelmentioning

confidence: 98%

“…The upshot is that this representation allows us to make use of techniques originally developed for the non-linear H 2|2 sigma model [20,21,[49][50][51] and the vertex-reinforced jump process [4,45]. In particular, our proof of Theorem 1.3 makes use of an adaptation of a Mermin-Wagner argument for the H 2|2 model [6,33,44]; the particular argument we adapt is due to Sabot [44]. For more on the connections between these models, see [6,45].…”

Section: Related Literaturementioning

confidence: 99%

“…In particular, our proof of Theorem 1.3 makes use of an adaptation of a Mermin-Wagner argument for the H 2|2 model [6,33,44]; the particular argument we adapt is due to Sabot [44]. For more on the connections between these models, see [6,45]. Conjecture 1.8 seems to have first appeared in print in [30].…”

Section: Related Literaturementioning

confidence: 99%

“…In contrast with the case of the complete graph, we show that on Z 2 the tree containing a fixed vertex always has finite density. Our arguments are closely based on estimates developed for the vertex-reinforced jump process [6,33,44]. The main new idea is to use these bounds in combination with dimensional reduction from Sect.…”

Section: No Percolation In Two Dimensionsmentioning

confidence: 99%

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Random Spanning Forests and Hyperbolic Symmetry

Bauerschmidt

Crawford

Helmuth

et al. 2020

Commun. Math. Phys.

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We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter $$\beta >0$$ β > 0 per edge. This is called the arboreal gas model, and the special case when $$\beta =1$$ β = 1 is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $$p=\beta /(1+\beta )$$ p = β / ( 1 + β ) conditioned to be acyclic, or as the limit $$q\rightarrow 0$$ q → 0 with $$p=\beta q$$ p = β q of the random cluster model. It is known that on the complete graph $$K_{N}$$ K N with $$\beta =\alpha /N$$ β = α / N there is a phase transition similar to that of the Erdős–Rényi random graph: a giant tree percolates for $$\alpha > 1$$ α > 1 and all trees have bounded size for $$\alpha <1$$ α < 1 . In contrast to this, by exploiting an exact relationship between the arboreal gas and a supersymmetric sigma model with hyperbolic target space, we show that the forest constraint is significant in two dimensions: trees do not percolate on $${\mathbb {Z}}^2$$ Z 2 for any finite $$\beta >0$$ β > 0 . This result is a consequence of a Mermin–Wagner theorem associated to the hyperbolic symmetry of the sigma model. Our proof makes use of two main ingredients: techniques previously developed for hyperbolic sigma models related to linearly reinforced random walks and a version of the principle of dimensional reduction.

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Section: The Arboreal Gas and Uniform Forest Modelmentioning

confidence: 98%

Section: Related Literaturementioning

confidence: 99%

Section: Related Literaturementioning

confidence: 99%

Section: No Percolation In Two Dimensionsmentioning

confidence: 99%

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Random Spanning Forests and Hyperbolic Symmetry

Bauerschmidt

Crawford

Helmuth

et al. 2020

Commun. Math. Phys.

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“…Since this falls under our Theorem 1.3 this gives a proof of recurrence of VRJP. We remark that a proof of a weaker notion of recurrence was given recently in [2].…”

Section: Exact Definitions and Statementsmentioning

confidence: 94%

Power-law decay of weights and recurrence of the two-dimensional VRJP

Kozma

Peled

2021

Electron. J. Probab.

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The vertex-reinforced jump process (VRJP) is a form of self-interacting random walk in which the walker is biased towards returning to previously visited vertices with the bias depending linearly on the local time at these vertices. We prove that, for any initial bias, the weights sampled from the magic formula on a two-dimensional graph decay at least at a power-law rate. Via arguments of Sabot and Zeng, the result implies that the VRJP is recurrent in two dimensions for any initial bias.

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$$\mathbb {H}^{2|2}$$-model and Vertex-Reinforced Jump Process on Regular Trees: Infinite-Order Transition and an Intermediate Phase

Poudevigne–Auboiron,

Wildemann

2024

Commun. Math. Phys.

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We explore the supercritical phase of the vertex-reinforced jump process (VRJP) and the $$\mathbb {H}^{2|2}$$ H 2 | 2 -model on rooted regular trees. The VRJP is a random walk, which is more likely to jump to vertices on which it has previously spent a lot of time. The $$\mathbb {H}^{2|2}$$ H 2 | 2 -model is a supersymmetric lattice spin model, originally introduced as a toy model for the Anderson transition. On infinite rooted regular trees, the VRJP undergoes a recurrence/transience transition controlled by an inverse temperature parameter $$\beta > 0$$ β > 0 . Approaching the critical point from the transient regime, $$\beta \searrow \beta _{\textrm{c}}$$ β ↘ β c , we show that the expected total time spent at the starting vertex diverges as $$\sim \exp (c/\sqrt{\beta - \beta _{\textrm{c}}})$$ ∼ exp ( c / β - β c ) . Moreover, on large finite trees we show that the VRJP exhibits an additional intermediate regime for parameter values $$\beta _{\textrm{c}}< \beta < \beta _{\textrm{c}}^{\textrm{erg}}$$ β c < β < β c erg . In this regime, despite being transient in infinite volume, the VRJP on finite trees spends an unusually long time at the starting vertex with high probability. We provide analogous results for correlation functions of the $$\mathbb {H}^{2|2}$$ H 2 | 2 -model. Our proofs rely on the application of branching random walk methods to a horospherical marginal of the $$\mathbb {H}^{2|2}$$ H 2 | 2 -model.

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Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process

Cited by 14 publications

References 33 publications

Random Spanning Forests and Hyperbolic Symmetry

Random Spanning Forests and Hyperbolic Symmetry

Power-law decay of weights and recurrence of the two-dimensional VRJP

$$\mathbb {H}^{2|2}$$-model and Vertex-Reinforced Jump Process on Regular Trees: Infinite-Order Transition and an Intermediate Phase

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