Fractal structure of zeros in hierarchical models

Derrida, Bernard; Sèze, L. De; Itzykson, C.

doi:10.1007/bf01018834

Cited by 183 publications

(152 citation statements)

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“…Previous studies of similar Ising models [9, 10] and Potts model [11] have not detected critical properties, due to the fact that infinite order transitions are elusive. On the contrary, a second order phase transition, as a function of the noise parameter, has been detected for a majority vote model [12].Ising models on different hierarchical fractal have been studied [13] and in the case of diamond fractal they have been exactly solved by exact renormalization [14,15]. These models, differently to present model, show a second order phase transition.…”

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confidence: 96%

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Ising models on the regularized Apollonian network

Serva¹,

Fulco²,

Albuquerque³

2013

Phys. Rev. E

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We investigate the critical properties of Ising models on a Regularized Apollonian Network (RAN), here defined as a kind of Apollonian Network (AN) in which the connectivity asymmetry associated to its corners is removed. Different choices for the coupling constants between nearest neighbors are considered, and two different order parameters are used to detect the critical behaviour. While ordinary ferromagnetic and anti-ferromagnetic models on RAN do not undergo a phase transition, some anti-ferrimagnetic models show an interesting infinite order transition. All results are obtained by an exact analytical approach based on iterative partial tracing of the Boltzmann factor as intermediate steps for the calculation of the partition function and the order parameters.PACS numbers: 89.75. Hc, 89.20.Hh, 89.75.Da Many real world networks exhibit complex topological properties as the small word effect, related to a very short minimal path between nodes, and the scale-free property, related to the power-law nature of the connectivity distribution. These properties have important implications in the real phenomena as virus spreading in computers, sharing of technological information and diffusion of epidemic diseases, to name just a few.In this context, the Apollonian network [1] is a particularly useful theoretical tool, since it is scale-free, displays small-world effect, can be embedded in a Euclidean lattice and shows space-filling as well as matching graph properties. Therefore, in spite of its deterministic nature, it shares the most relevant characteristics of real world networks.Phase transitions has been detected for a number of different physical models on Apollonian Networks. For example, the ideal gas undergoes to Bose-Einstein condensation [2-6] and epidemics exhibits a transition between an absorbing state and an active state [7,8]. In particular, in [6] it has been adopted an analytical strategy which has some similarities with that in this paper.In this work we focus on the infinite order transition exhibited by some Ising models on the Apollonian network. Previous studies of similar Ising models [9, 10] and Potts model [11] have not detected critical properties, due to the fact that infinite order transitions are elusive. On the contrary, a second order phase transition, as a function of the noise parameter, has been detected for a majority vote model [12].Ising models on different hierarchical fractal have been studied [13] and in the case of diamond fractal they have been exactly solved by exact renormalization [14,15]. These models, differently to present model, show a second order phase transition. Indeed, the standard Ising behavior is second order transition in plane models [16,17], nevertheless, it may be very intricate, with many phases [18], when the interactions are made more complicated.We start by regularizing the standard Apollonian Network in order to remove the connectivity asymmetry associated to its corners which consistently simplify the analytical computation of the thermodynamics o...

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confidence: 96%

“…Ising models on different hierarchical fractal have been studied [13] and in the case of diamond fractal they have been exactly solved by exact renormalization [14,15]. These models, differently to present model, show a second order phase transition.…”

mentioning

confidence: 99%

Ising models on the regularized Apollonian network

Serva¹,

Fulco²,

Albuquerque³

2013

Phys. Rev. E

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“…The wetting problem on a hierarchical lattice. One of the developments in the theory of critical phenomena by the renormalization group approach was to study statistical mechanical models on hierarchical lattices [4,28,23,15,31]. These lattices are usually constructed by a recursive procedure which is at the origin of a discrete scale invariance and which allows one to write exact renormalization transformations.…”

Section: Two Examplesmentioning

confidence: 99%

Log-periodic Critical Amplitudes: A Perturbative Approach

Derrida

Giacomin

2013

J Stat Phys

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Abstract. Log-periodic amplitudes appear in the critical behavior of a large class of systems, in particular when a discrete scale invariance is present. Here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial. In this case, the log-periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated.Dedicated to Herbert Spohn, our colleague, friend and source of inspiration, on the occasion of his 65 th birthday

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“…Models such as this are interesting in studies of the roots of partition functions [25,26], since the computation of large numbers of zeros for such partition functions is easily accomplished.…”

Section: Which Fixes Q (A)mentioning

confidence: 99%

Nuclear fragmentation and its parallels

Kc¹,

Az²

1994

Phys. Rev. C

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A model for the fragmentation of a nucleus is developed. Parallels of the description of this process with other areas are shown which include Feynman's theory of the λ transition in liquid Helium, Bose condensation, and Markov process models used in stochastic networks and polymer physics. These parallels are used to generalize and further develop a previous exactly solvable model of nuclear fragmentation. An analysis of some experimental data is given.

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Fractal structure of zeros in hierarchical models

Cited by 183 publications

References 14 publications

Ising models on the regularized Apollonian network

Ising models on the regularized Apollonian network

Log-periodic Critical Amplitudes: A Perturbative Approach

Nuclear fragmentation and its parallels

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