Local convergence of large random triangulations coupled with an Ising model

Albenque, Marie; Ménard, Laurent; Schaeffer, Gilles

doi:10.1090/tran/8150

Cited by 8 publications

(33 citation statements)

References 85 publications

(157 reference statements)

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“…Thus, x(z) and y(z) can be put in the form (18), the first condition (60) implying that b m−1 = 1. By extracting the coefficient of z 1−km for k 1 in the second condition (60), we find that α k = a mk−1 is given by ( 4), while extracting the coefficient of z yields an equation equivalent to (3). Identifying F • p =W mp and F • p = W mp , we obtain Propositions 2 and 3 as specializations of Theorem 3.…”

Section: B Rational Parametrizations For Monochromatic Boundariesmentioning

confidence: 88%

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Enumeration of Planar Constellations with an Alternating Boundary

Bouttier¹,

Carrance

2021

Electron. J. Combin.

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A planar hypermap with a boundary is defined as a planar map with a boundary, endowed with a proper bicoloring of the inner faces. The boundary is said alternating if the colors of the incident inner faces alternate along its contour. In this paper we consider the problem of counting planar hypermaps with an alternating boundary, according to the perimeter and to the degree distribution of inner faces of each color. The problem is translated into a functional equation with a catalytic variable determining the corresponding generating function. In the case of constellations—hypermaps whose all inner faces of a given color have degree $m\geq 2$, and whose all other inner faces have a degree multiple of $m$—we completely solve the functional equation, and show that the generating function is algebraic and admits an explicit rational parametrization. We finally specialize to the case of Eulerian triangulations—hypermaps whose all inner faces have degree $3$—and compute asymptotics which are needed in another work by the second author, to prove the convergence of rescaled planar Eulerian triangulations to the Brownian map.

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Section: B Rational Parametrizations For Monochromatic Boundariesmentioning

confidence: 88%

“…A nice account of the enumerative results coming from this approach may be found in the last chapter of the book by Eynard [21]. Let us also mention the many recent papers [3,17,18,27] devoted to the local limits of the Ising model on random maps.…”

Section: Introductionmentioning

confidence: 99%

Enumeration of Planar Constellations with an Alternating Boundary

Bouttier¹,

Carrance

2021

Electron. J. Combin.

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“…In another article [2], we derive several critical exponents for the sign clusters in finite and infinite planar triangulations coupled with an Ising model. In particular, we establish in [2] counterparts for the main results of the present work. Theorem 1 and its counterpart for Ising model are unrelated, but Theorem 2 is a particular case of its Ising version at infinite temperature.…”

Section: Theorem 1 Let P Pmentioning

confidence: 99%

“…The stationary points of ŷ(p, U (t 3 ), V ) are the roots of the polynomial 2 , where U stands for U (t 3 ). By computing the values of the polyomial at 0, 1 − 2U and 2(1 − 2U ), we can see that it has 4 real roots…”

Section: Lemma 4 Recall the Definition Of The Power Seriesmentioning

confidence: 99%

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Percolation probability and critical exponents for site percolation on the UIPT

Ménard¹

2022

Preprint

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We derive three critical exponents for Bernoulli site percolation on the on the Uniform Infinite Planar Triangulation (UIPT). First we compute explicitly the probability that the root cluster is infinite. As a consequence, we show that the off-critical exponent for site percolation on the UIPT is β = 1/2. Then we establish an integral formula for the generating function of the number of vertices in the root cluster. We use this formula to prove that, at criticality, the probability that the root cluster has at least n vertices decays like n −1/7 . Finally, we also derive an expression for the law of the perimeter of the root cluster and use it to establish that, at criticality, the probability that the perimeter of the root cluster is equal to n decays like n −4/3 . Among these three exponents, only the last one was previously known. Our main tools are the so-called gasket decomposition of percolation clusters, generic properties of random Boltzmann maps, as well as analytic combinatorics.

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Critical Ising Model on Random Triangulations of the Disk: Enumeration and Local Limits

Chen

Turunen

2020

Commun. Math. Phys.

134

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In [ ], we have studied the Boltzmann random triangulation of the disk coupled to an Ising model with Dobrushin boundary condition at its critical temperature. In this paper, we investigate the phase transition of this model by extending our previous results to arbitrary temperature: We compute the partition function of the model at all temperatures, and derive several critical exponents associated with the in nite perimeter limit. We show that the model has a local limit at any temperature, whose properties depend drastically on the temperature. At high temperatures, the local limit is reminiscent of the uniform in nite half-planar triangulation (UIHPT) decorated with a subcritical percolation. At low temperatures, the local limit develops a bottleneck of nite width due to the energy cost of the main Ising interface between the two spin clusters imposed by the Dobrushin boundary condition. This change can be summarized by a novel order parameter with a nice geometric meaning. In addition to the phase transition, we also generalize our construction of the local limit from the two-step asymptotic regime used in [ ] to a more natural diagonal asymptotic regime. We obtain in this regime a scaling limit related to the length of the main Ising interface, which coincides with predictions from the continuum theory of quantum surfaces (a.k.a. Liouville quantum gravity). their roots in the physics literature [ ], and was revisited and popularized by Angel in [ ]. The peeling process proves to be a valuable tool for understanding the geometry of random planar maps without Ising model, see [ ] for a review of recent developments.In a previous paper [ ], we extended some enumeration results of Bernardi and Bousquet-Mélou [ ] to study the Ising-decorated random triangulations with Dobrushin boundary condition at its critical temperature. We used the peeling process to construct the local limit of the model, and to obtain several scaling limit results concerning the lengths of some Ising interfaces. In this paper, we extend similar results to the model at any temperature, and show how the large scale geometry of Ising-decorated random triangulations changes qualitatively at the critical temperature. In particular, our results con rm the physical intuition that, at large scale, Ising-decorated random maps at non-critical temperatures behave like non-decorated random maps.A similar model of Ising-decorated triangulations has been studied in a recent work of Albenque, Ménard and Schae er [ ]. They followed an approach reminiscent of Angel and Schramm in [ ] to show that the model has a local limit at any temperature, and obtained several properties of the limit such as one-endedness and recurrence for some temperatures. However they studied the model without boundary, and hence did not encounter the geometric consequences of the phase transition.We start by recalling some essential de nitions from [ ].Planar maps. Recall that a ( nite) planar map is a proper embedding of a nite connected graph into the sphere S 2 , viewed up to o...

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Local convergence of large random triangulations coupled with an Ising model

Cited by 8 publications

References 85 publications

Enumeration of Planar Constellations with an Alternating Boundary

Enumeration of Planar Constellations with an Alternating Boundary

Percolation probability and critical exponents for site percolation on the UIPT

Critical Ising Model on Random Triangulations of the Disk: Enumeration and Local Limits

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