One step replica symmetry breaking and overlaps between two temperatures

Derrida, Bernard; Mottishaw, Peter

doi:10.1088/1751-8121/abd4ad

Cited by 7 publications

(6 citation statements)

References 38 publications

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“…It is not too difficult to see that E[ρ 1,2 ] is asymptotic to the inverse coalescent time, c ζ : First, notice that conditional on {Φ i } i ] the distribution of ϱ 1,2 is (see [DM21])…”

Section: Gibbs Measurementioning

confidence: 99%

Coalescent processes emerging from large deviations

Levien

2024

J. Stat. Mech.

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The classical model for the genealogies of a neutrally evolving population in a fixed environment is due to Kingman. Kingman’s coalescent process, which produces a binary tree, emerges universally from many microscopic models in which the variance in the number of offspring is finite. It is understood that power-law offsprings distributions with infinite variance can result in a very different type of coalescent structure with merging of more than two lineages. Here, we investigate the regime where the variance of the offspring distribution is finite but comparable to the population size. This is achieved by studying a model in which the log offspring sizes have stretched exponential tails. Such offspring distributions are motivated by biology, where they emerge from a toy model of growth in a heterogeneous environment, but also from mathematics and statistical physics, where limit theorems and phase transitions for sums over random exponentials have received considerable attention due to their appearance in the partition function of Derrida’s random energy model (REM). We find that the limit coalescent is a β-coalescent—a previously studied model emerging from evolutionary dynamics models with heavy-tailed offspring distributions. We also discuss the connection to previous results on the REM.

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Section: Gibbs Measurementioning

confidence: 99%

Coalescent processes emerging from large deviations

Levien

2024

J. Stat. Mech.

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“…In a very recent article, Derrida and Mottishaw [13] gave a thorough analysis of the overlap between two copies of the same REM at two temperatures. The distribution of the overlap between two points sampled independently according to Gibbs measures at temperatures β and β ′ is in our settings G β,t ⊗ G β ′ ,t (q t (u, v) ∈ •).…”

Section: Overviewmentioning

confidence: 99%

The overlap distribution at two temperatures for the branching Brownian motion

Bonnefont¹

2022

Preprint

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We study the overlap distribution of two particles chosen under the Gibbs measure at two temperatures for the branching Brownian motion. We first prove the convergence of the overlap distribution using the extended convergence of the extremal process obtained by Bovier and Hartung [8]. We then prove that the mean overlap of two points chosen at different temperatures is strictly smaller than in Derrida's random energy model. The proof of this last result is achieved with the description of the decoration point process obtained by Aïdékon, Berestycki, Brunet and Shi [1]. To our knowledge, it is the first time that this description is being used.

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“…In a very recent article, Derrida and Mottishaw [14] gave a thorough analysis of the overlap between two copies of the same REM at two temperatures. The distribution of the overlap between two points sampled independently according to Gibbs measures at temperatures β and β is in our settings G β,t ⊗ G β ,t (q t (u, v) ∈ •).…”

Section: Overviewmentioning

confidence: 99%