Energy Correlations of Non-Integrable Ising Models: The Scaling Limit in the Cylinder

Antinucci, Giovanni; Giuliani, Alessandro; Greenblatt, Rafael L.

doi:10.1007/s00220-022-04481-z

Cited by 4 publications

(1 citation statement)

References 74 publications

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“…Introductions to fermionic renormalisation include [22,66,73], see also [53]. Recent probabilistic applications of these approaches to fermionic renormalisation include the study of interacting dimers [51,52] and two-dimensional finite range Ising models [7,8,49,50]. Our organisation of the renormalisation group is instead based on a finite range decomposition and polymer coordinates, and follows [28] and its further developments in [12,16,17,[29][30][31][32]36].…”

Section: Background On Non-linear Sigma Models and Renormalisationmentioning

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: Background On Non-linear Sigma Models and Renormalisationmentioning

confidence: 99%

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

Bauerschmidt,

Crawford,

Helmuth

2024

Invent. math.

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The surface counter-terms of the ϕ44 theory on the half space R+×R3

Borji,

Kopper

2024

Journal of Mathematical Physics

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In a previous work, we established perturbative renormalizability to all orders of the massive ϕ44-theory on a half-space also called the semi-infinite massive ϕ44-theory. Five counter-terms which are functions depending on the position in the space, were needed to make the theory finite. The aim of the present paper is to establish that for a particular choice of the renormalization conditions the effective action consists of a part which is independent of the boundary conditions (Dirichlet, Neumann and Robin) plus a boundary term in the case of the Robin and Neumann boundary conditions. The key idea of our method is the decomposition of the correlators into a bulk part, which is defined as the scalar field model on the full space R4 with a quartic interaction restricted to the half-space, plus a remainder which we call “the surface part.” We analyse this surface part and establish perturbatively that the ϕ44 theory in R+×R3 is made finite by adding the bulk counter-terms and two additional counter-terms to the bare interaction in the case of Robin and Neumann boundary conditions. These surface counter-terms are position independent and are proportional to ∫Sϕ2 and ∫Sϕ∂nϕ. For Dirichlet boundary conditions, we prove that no surface counter-terms are needed and the bulk counter-terms are sufficient to renormalize the connected amputated (Dirichlet) Schwinger functions. A key technical novelty as compared to our previous work is a proof that the power counting of the surface part of the correlators is better by one scaling dimension than their bulk counterparts.

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