To study the depinning transition in the limit of strong disorder, Derrida and Retaux [9] introduced a discrete-time max-type recursive model. It is believed that for a large class of recursive models, including Derrida and Retaux' model, there is a highly non-trivial phase transition. In this article, we present a continuous-time version of Derrida and Retaux model, built on a Yule tree, which yields an exactly solvable model belonging to this universality class. The integrability of this model allows us to study in details the phase transition near criticality and can be used to confirm the infinite order phase transition predicted by physicists. We also study the scaling limit of this model at criticality, which we believe to be universal. partially supported by ANR MALIN arXiv:1811.08749v2 [math.PR] 1 Apr 2019Note that the DR process can be seen as a discrete-time version of a solution of a McKean-Vlasov type SDE, see McKean [20], i.e. a Markov process interacting with its distribution. It follows from an easy recursion that for all n ∈ N, X n has law µ n .The aim of this article is to define continuous-time versions of the DR model, tree and process. In particular, we are looking for a process with a density r t (x) with respect to the Lebesgue measure which could solve [9, Equation (33)], that we recall here(1.8)This differential equation (1.8) plays a key role in the prediction (1.5), and was obtained by Derrida and Retaux as an informal scaling limit of the sequence of measures (µ nt (ndx)).

Consider the supercritical branching random walk on the real line in the boundary case and the associated Gibbs measure ν n,β on the n th generation, which is also the polymer measure on a disordered tree with inverse temperature β. The convergence of the partition function W n,β , after rescaling, towards a nontrivial limit has been proved by Aïdékon and Shi [3] in the critical case β = 1 and by Madaule [34] when β > 1. We study here the near-critical case, where β n → 1, and prove the convergence of W n,βn , after rescaling, towards a constant multiple of the limit of the derivative martingale. Moreover, trajectories of particles chosen according to the Gibbs measure ν n,β have been studied by Madaule [35] in the critical case, with convergence towards the Brownian meander, and by Chen, Madaule and Mallein [22] in the strong disorder regime, with convergence towards the normalized Brownian excursion. We prove here the convergence for trajectories of particles chosen according to the nearcritical Gibbs measure and display continuous families of processes from the meander to the excursion or to the Brownian motion.MSC 2010: 60J80, 60F05, 60F17.According to Derrida and Spohn [24], there is a critical parameter β c > 0 for the directed polymer on a disordered tree (with our setting β c = 1, see Subsection 1.2 for more details) and our aim in this paper is to study the near-critical case, where β depends on n and tends from above and below to β c = 1 as n → ∞. The near-critical case has been recently studied for the directed polymer on the lattice in dimension 1 + 1 and 1 + 2 by Alberts, Khanin and Quastel [4] and Caravenna, Sun and Zygouras [17,18], with the emergence of the so-called intermediate disorder regime. For the polymer on a tree, some work near criticality has been done by Alberts and Ortgiese [5] and Madaule [35], mostly on the partition function.Before stating our results, we recall some well-known properties of the branching random walk, that hold under assumptions (1.1), (1.2) and (1.3). First, the sequenceis a martingale, called the derivative martingale, and Biggins and Kyprianou [11] (under slightly stronger assumptions) and Aïdékon [1] showed that we have( 1.7) Moreover, Chen [20] proved that these assumptions are optimal for the nontriviality of D ∞ . Furthermore, Aïdékon [1] also showed that, in the nonlattice case, min |z|=n V (z) − 3 2 log n converges in law under P * and described the limit as a random shift (depending on D ∞ ) of a Gumbel distribution. In the lattice case, we do not have this convergence, but the tightness still holds (see Equation (4.20) of Chen [21] or Mallein [36]): for each ε > 0, it exists C > 0 such thatfor n large enough. The partition functionThe process (W n,β ) n∈N for some fixed β ∈ R + has been intensively studied because, if Ψ(β) is finite, then the renormalized process ( W n,β ) n∈N := (e −nΨ(β) W n,β ) n∈N is a nonnegative martingale, called additive martingale, and, therefore, converges a.s. to some limit W ∞,β . Kahane and Peyrière [31], Biggins [7] and...

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