The critical behaviors and the scaling functions of a coalescence equation

We consider a recursive system (X n ) which was introduced by Collet et al. [10]) as a spin glass model, and later by Derrida, Hakim, and Vannimenus [13] and by Derrida and Retaux [14] as a simplified hierarchical renormalization model. The system (X n ) is expected to possess highly nontrivial universalities at or near criticality. In the nearly supercritical regime, Derrida and Retaux [14] conjectured that the free energy of the system decays exponentially with exponent (p − p c ) − 1 2 as p ↓ p c . We study the nearly subcritical regime (p ↑ p c ) and aim at a dual version of the Derrida-Retaux conjecture; our main result states that as n → ∞, both E(X n ) and P(X n = 0) decay exponentially with exponent (p c − p) 1 2 +o(1) , where o(1) → 0 as p ↑ p c .

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“…Let ε 0 ∈ (0, pc 2 ) be small (how small will be determined later) and ε ∈ (0, ε 0 ). By Lemma 2.6, we have 5 (2.16)…”

Section: Proof Of Proposition 21mentioning

confidence: 94%

“…Curien and Hénard [12], Contat and Curien [11], and Aldous et al [2]. See [19] for an extension to the case when m is random, and [18,5] for an exactly solvable version in continuous time.…”

Section: Introductionmentioning

confidence: 99%

The Derrida–Retaux conjecture on recursive models

et al. 2021

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“…Let us also mention that some continuous-time systems were studied in [12], [19] and [5], for which the analogue of the asymptotic equivalences of P(Y n ≥ 1) was proved; see [12] and [19] for the analogue of (1.4), and [5] for the analogue of (1.5).…”

mentioning

confidence: 99%

The sustainability probability for the critical Derrida-Retaux model

Chen¹,

Hu²,

Shi³

2021

Preprint

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We are interested in the recursive model (Y n , n ≥ 0) studied by Collet, Eckmann, Glaser and Martin [9] and by Derrida and Retaux [12]. We prove that at criticality, the probability P(Y n > 0) behaves like n −2+o(1) as n goes to infinity; this gives a weaker confirmation of predictions made in [9], [12] and [6]. Our method relies on studying the number of pivotal vertices and open paths, combined with a delicate coupling argument.

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$$\mathbb {H}^{2|2}$$-model and Vertex-Reinforced Jump Process on Regular Trees: Infinite-Order Transition and an Intermediate Phase

Poudevigne–Auboiron,

Wildemann

2024

Commun. Math. Phys.

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We explore the supercritical phase of the vertex-reinforced jump process (VRJP) and the $$\mathbb {H}^{2|2}$$ H 2 | 2 -model on rooted regular trees. The VRJP is a random walk, which is more likely to jump to vertices on which it has previously spent a lot of time. The $$\mathbb {H}^{2|2}$$ H 2 | 2 -model is a supersymmetric lattice spin model, originally introduced as a toy model for the Anderson transition. On infinite rooted regular trees, the VRJP undergoes a recurrence/transience transition controlled by an inverse temperature parameter $$\beta > 0$$ β > 0 . Approaching the critical point from the transient regime, $$\beta \searrow \beta _{\textrm{c}}$$ β ↘ β c , we show that the expected total time spent at the starting vertex diverges as $$\sim \exp (c/\sqrt{\beta - \beta _{\textrm{c}}})$$ ∼ exp ( c / β - β c ) . Moreover, on large finite trees we show that the VRJP exhibits an additional intermediate regime for parameter values $$\beta _{\textrm{c}}< \beta < \beta _{\textrm{c}}^{\textrm{erg}}$$ β c < β < β c erg . In this regime, despite being transient in infinite volume, the VRJP on finite trees spends an unusually long time at the starting vertex with high probability. We provide analogous results for correlation functions of the $$\mathbb {H}^{2|2}$$ H 2 | 2 -model. Our proofs rely on the application of branching random walk methods to a horospherical marginal of the $$\mathbb {H}^{2|2}$$ H 2 | 2 -model.

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The critical behaviors and the scaling functions of a coalescence equation

Cited by 7 publications

References 24 publications

The Derrida–Retaux conjecture on recursive models

The Derrida–Retaux conjecture on recursive models

The sustainability probability for the critical Derrida-Retaux model

$$\mathbb {H}^{2|2}$$-model and Vertex-Reinforced Jump Process on Regular Trees: Infinite-Order Transition and an Intermediate Phase

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