The Derrida–Retaux conjecture on recursive models

Abstract

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 .

The sustainability probability for the critical Derrida-Retaux model

“…), so the second inequality in (ii) follows from (i); for (iii) and (iv), see [4] and [6], respectively.…”

“…The system is known for exhibiting a phase transition: 4 Theorem A. (Collet et al [9]) Assume E(X 0 m X 0 ) < ∞.…”

“…with c * := 4 (m−1) 2 and c * * := m m−1 c * . In [4], it was realized that (1.4) should be valid only under the assumption E(Y 3 0 m Y 0 ) < ∞. When the latter condition fails, P(Y n ≥ 1) and E(Y n ) should behave differently.…”

“…When the latter condition fails, P(Y n ≥ 1) and E(Y n ) should behave differently. For example, for m = 2, if P(Y 0 = k) ∼ c 0 k −α 2 −k , k → ∞, for some α ∈ (2,4] (so E(Y 3 0 m Y 0 ) = ∞) and c 0 ∈ (0, ∞), then it is expected (see [13]) that (1.5)…”

“…The notion of open paths was already introduced in [4], and was known to be an important tool in the study of the Derrida-Retaux model. From methodological point of view, an interesting contribution of the paper is to introduce the notion of pivotal vertices, and to prove Theorem 2.2 on the expected number of pivotal vertices.…”