Pathologies of the large-N limit for RPN−1, CPN−1, QPN−1 and mixed isovector/isotensor σ-models

Sokal, Alan D.; Starinets, Andrei O.

doi:10.1016/s0550-3213(01)00065-7

Cited by 40 publications

(51 citation statements)

References 107 publications

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“…Recall from Section 2.8 the Mellin inversion formula, (25). Let us set x = y = 0 and move the contour of integration to the line Re(σ) = −1/4.…”

Section: The Lead Term When D =mentioning

confidence: 99%

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Chinta¹,

Jorgenson²

2010

Nagoya Mathematical Journal

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Abstract. By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of I-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz-type zeta functions.

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“…Recall from Section 2.8 the Mellin inversion formula, (25). Let us set x = y = 0 and move the contour of integration to the line Re(σ) = −1/4.…”

Section: The Lead Term When D =mentioning

confidence: 99%

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Chinta¹,

Jorgenson²

2010

Nagoya Mathematical Journal

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“…constitutes the starting point for canonical thermodynamics. An exact expression for f (β) can be obtained in various ways: either by a large N-expansion (18) of the integral in (4.3), or, analogously to the calculations of Berlin and Kac, (14) by making use of the method of steepest descent. Having already at hand an expression for the microcanonical entropy density s, the free energy density is most easily obtained by means of a Legendre-Fenchel transform,…”

Section: Canonical Solutionmentioning

confidence: 99%

“…(26) For the mean-field spherical model, a discussion of the Fisher zeros of the zero-field canonical partition function (4.3) amounts to analyzing the complex zeros of the confluent hypergeometric function 1 F 1 ( 1 2 , N 2 , ·) with respect to its third argument. Focusing on the large N asymptotics, we are interested in the complex zeros of 1 F 1 ( 1 2 , N 2 , N β 2 ) with respect to β in the limit of large N. At this point, we can benefit from the work by Sokal and Starinets: (18) They proved that, as illustrated in figure 4, the complex zeros lie close to the two branches with real part ℜ(β) > 1 of the Szegö curve |β| 2 = e 2(ℜ(β)−1) , (6.1) approaching the curve and the value β = 1 in the thermodynamic limit of large N. This is a beautiful illustration of the Yang-Lee-mechanism of analyticity breaking, in which, for large but finite N, the complex zeros of a real-analytic function (the canonical partition function) approach and pinch the real axis, giving rise to a nonanalyticity in the limit N → ∞. More precisely, in this limit the zeros lie dense on the branches of the Szegö curve, thus cutting the complex plane into two disconnected domains.…”

Section: Fisher Zeros Of the Canonical Partition Functionmentioning

confidence: 99%

On the Mean-Field Spherical Model

Kastner

Schnetz

2006

J Stat Phys

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Exact solutions are obtained for the mean-field spherical model, with or without an external magnetic field, for any finite or infinite number N of degrees of freedom, both in the microcanonical and in the canonical ensemble. The canonical result allows for an exact discussion of the loci of the Fisher zeros of the canonical partition function. The microcanonical entropy is found to be nonanalytic for arbitrary finite N . The mean-field spherical model of finite size N is shown to be equivalent to a mixed isovector/isotensor σ-model on a lattice of two sites. Partial equivalence of statistical ensembles is observed for the mean-field spherical model in the thermodynamic limit. A discussion of the topology of certain state space submanifolds yields insights into the relation of these topological quantities to the thermodynamic behavior of the system in the presence of ensemble nonequivalence.Key Words: Phase transitions, spherical model, microcanonical, canonical, ensemble nonequivalence, partial equivalence, Fisher zeros of the partition function, ÊÈ N −1 σ-model, mixed isovector/isotensor σ-model, model equivalence, topological approach.

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“…The constant c d is the same as for the spanning trees and it is known that c 1 = 0 and c 2 = 4G π , where G is Catalan's constant (see [2] or [17]). The difference lies in the second term (if we forget about the log(u 2 ) term in [2]).…”

Section: Introductionmentioning

confidence: 99%

The bundle Laplacian on discrete tori

Friedli

2018

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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We prove an asymptotic formula for the determinant of the bundle Laplacian on discrete d-dimensional tori as the number of vertices tends to infinity. This determinant has a combinatorial interpretation in terms of cycle-rooted spanning forests. We also establish a relation (in the limit) between the spectral zeta function of a line bundle over a discrete torus, the spectral zeta function of the infinite graph Z d and the Epstein-Hurwitz zeta function. The latter can be viewed as the spectral zeta function of the twisted continuous torus which is the limit of the sequence of discrete tori.

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Pathologies of the large-N limit for RPN−1, CPN−1, QPN−1 and mixed isovector/isotensor σ-models

Cited by 40 publications

References 107 publications

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

On the Mean-Field Spherical Model

The bundle Laplacian on discrete tori

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