Generic Ising trees

Durhuus, Bergfinnur; Napolitano, George

doi:10.1088/1751-8113/45/18/185004

“…Hence, the coupling of the dense loop model to CDT does not influence the statistical behaviour of the underlying triangulations. This is analogous to what is seen for the Ising model coupled to a random planar tree, where a relation similar to (5.2) can be derived [39].…”

Section: Dense Loop Modelsupporting

confidence: 78%

Critical behaviour of loop models on causal triangulations

Durhuus¹,

Poncini²,

Rasmussen³

et al. 2021

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We introduce a dense and a dilute loop model on causal dynamical triangulations. Both models are characterised by a geometric coupling constant g and a loop parameter α in such a way that the purely geometric causal triangulation model is recovered for α = 1. We show that the dense loop model can be mapped to a solvable planar tree model, whose partition function we compute explicitly and use to determine the critical behaviour of the loop model. The dilute loop model can likewise be mapped to a planar tree model; however, a closed-form expression for the corresponding partition function is not obtainable using the standard methods employed in the dense case. Instead, we derive bounds on the critical coupling g c and apply transfer matrix techniques to examine the critical behaviour for α small.

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“…Hence, the coupling of the dense loop model to CDT does not influence the statistical behaviour of the underlying triangulations. This is analogous to what is seen for the Ising model coupled to a random planar tree, where a relation similar to (5.2) can be derived [37].…”

Section: Dense Loop Modelsupporting

confidence: 78%

Critical behaviour of loop models on causal triangulations

Durhuus,

Poncini,

Rasmussen

et al. 2021

Preprint

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We introduce a dense and a dilute loop model on causal dynamical triangulations. Both models are characterised by a geometric coupling constant g and a loop parameter α in such a way that the purely geometric causal triangulation model is recovered for α = 1. We show that the dense loop model can be mapped to a solvable planar tree model, whose partition function we compute explicitly and use to determine the critical behaviour of the loop model. The dilute loop model can likewise be mapped to a planar tree model; however, a closed-form expression for the corresponding partition function is not obtainable using the standard methods employed in the dense case. Instead, we derive bounds on the critical coupling g c and apply transfer matrix techniques to examine the critical behaviour for α small.

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“…We mention some former works about the Ising model on random graphs. The Ising model on random trees is investigated in [16]. The Ising model on random surfaces is solvable in terms of the matrix model [17,18].…”

Section: Introductionmentioning

confidence: 99%

Ising model on random networks and the canonical tensor model

Sasakura

¹

,

Sato

²

2014

Progress of Theoretical and Experimental Physics

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We introduce a statistical system on random networks of trivalent vertices for the purpose of studying the canonical tensor model, which is a rank-three tensor model in the canonical formalism. The partition function of the statistical system has a concise expression in terms of integrals, and has the same symmetries as the kinematical ones of the canonical tensor model. We consider the simplest nontrivial case of the statistical system corresponding to the Ising model on random networks, and find that its phase diagram agrees with what is implied by regrading the Hamiltonian vector field of the canonical tensor model with N = 2 as a renormalization group flow. Along the way, we obtain an explicit exact expression of the free energy of the Ising model on random networks in the thermodynamic limit by the Laplace method. This paper provides a new example connecting a model of quantum gravity and a random statistical system. * sasakura@yukawa.kyoto-u.ac.jp † Yuki.Sato@wits.ac.za * Namely, the indices can take N values, say 1, 2, . . . , N .

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Generic Ising trees

Cited by 4 publications

References 28 publications

Critical behaviour of loop models on causal triangulations

Critical behaviour of loop models on causal triangulations

Critical behaviour of loop models on causal triangulations

Ising model on random networks and the canonical tensor model

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