Replica symmetry breaking in the random replicant model

Biscari, Paolo; Parisi, Giorgio

doi:10.1088/0305-4470/28/17/006

Cited by 35 publications

(52 citation statements)

References 11 publications

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“…We note that for Γ = 1 both the instability of the fixed point and the onset of memory occur at the same value of u = 1/ √ 2. In the statics a de Almeida-Thouless instability is found at this point and replica symmetry is broken at smaller values of u [16].…”

Section: The Case R = 2 P =mentioning

confidence: 83%

“…These describe the evolution of selfreproducing interacting species within a given framework of limited resources and have found wide applications in a variety of fields including game theory, socio-biology, prebiotic evolution and optimization theory [4,5,6]. While the first replicator system with quenched random couplings was introduced by Diederich and Opper in [7,8], most subsequent studies in the statistical physics community seem to be based on replica theory or on computer simulations [10,11,12,13,14,15,16,17,18]. Thus most of the existing analytic work on RRM is restricted to the case of symmetric couplings, in which a Lyapunov function can be found, so that replica theory is applicable.…”

Section: Introductionmentioning

confidence: 99%

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Random replicators with asymmetric couplings

Galla¹

2006

J. Phys. A: Math. Gen.

103

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Abstract. The dynamics of random replicators is studied using generating functional techniques. While replica analyses are limited to models with symmetric couplings, dynamical approaches as presented here allow specifically to address cases with asymmetric interactions. We first discuss in detail how dynamical theories can be formulated for general replicator models in terms of an effective single-species process, and how persistent order parameters of the ergodic stationary states can be extracted from this process. We then detail how different types of dynamical phase transitions can be identified and related to each other. As an application of the general theory we address replicator models with Gaussian couplings of arbitrary symmetry between pairs and triples of species, respectively. Numerical simulations verify our theory, and also indicate regimes in which only a finite number of species survives, even when the thermodynamic limit is considered.

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Section: The Case R = 2 P =mentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Random replicators with asymmetric couplings

Galla¹

2006

J. Phys. A: Math. Gen.

103

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“…The randomness of the interspecies couplings here reflects an amount of uncertainty about the structure of interactions in real ecosystems. The initial RRM and various extensions have subsequently been studied in a series of papers [8,9,10,11,12,13,14,15,16,17] and they have been found to exhibit intriguing features, both from the biological point of view as well as from the perspective of statistical mechanics. In particular it has been realised that the model exhibits phase behaviour with interesting ergodic and nonergodic phases, different types of phase transitions as well as replica-symmetry breaking.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of random replicators with Hebbian interactions

Galla¹

2005

J. Stat. Mech.

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Abstract. A system of replicators with Hebbian random couplings is studied using dynamical methods. The self-reproducing species are here characterized by a set of binary traits and interact based on complementarity. In the case of an extensive number of traits we use path-integral techniques to demonstrate how the coupled dynamics of the system can be formulated in terms of an effective single-species process in the thermodynamic limit, and how persistent order parameters characterizing the stationary states may be computed from this process in agreement with existing replica studies of the statics. Numerical simulations confirm these results. The analysis of the dynamics allows an interpretation of two different types of phase transitions of the model in terms of memory onset at finite and diverging integrated response, respectively. We extend the analysis to the case of diluted couplings of an arbitrary symmetry, where replica theory is not applicable. Finally the dynamics and in particular the approach to the stationary state of the model with a finite number of traits is addressed.

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“…In this letter we solve analytically a model of co-evolution of N species interacting via random, high-order interactions. Our model is a generalization of the random replicant model [6][7][8] which considers only pair interactions between the species.…”

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confidence: 99%

Random Replicators with High-Order Interactions

Oliveira

Fontanari

2000

Phys. Rev. Lett.

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We use tools of the equilibrium statistical mechanics of disordered systems to study analytically the statistical properties of an ecosystem composed of N species interacting via random, Gaussian interactions of order p ≥ 2, and deterministic self-interactions u ≥ 0. We show that for nonzero u the effect of increasing the order of the interactions is to make the system more cooperative, in the sense that the fraction of extinct species is greatly reduced.Furthermore, we find that for p > 2 there is a threshold value which gives a lower bound to the concentration of the surviving species, preventing then the existence of rare species and, consequently, increasing the robustness of the ecosystem to external perturbations.

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Replica symmetry breaking in the random replicant model

Cited by 35 publications

References 11 publications

Random replicators with asymmetric couplings

Random replicators with asymmetric couplings

Dynamics of random replicators with Hebbian interactions

Random Replicators with High-Order Interactions

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