The near-critical Gibbs measure of the branching random walk

Abstract

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...

On the trajectory of an individual chosen according to supercritical Gibbs measure in the branching random walk

“…When individuals are sampled according to ν n,β with β < 1, Chauvin and Rouault [11] proved that MRCA(z, z ′ ) converges in law, thus ρ β = 0 for β < 1. This result was then extended by Pain [29], who proved the same convergence holds when considering ν n,βn with β n → 1. The conjecture of Derrida and Spohn was partially proved by Bovier and Kurkova [7] for some binary branching processes with Gaussian increments, by Arguin and Zindy [3] for the overlapping probability in the 2-dimensional discrete Gaussian free field, and by Jagannath [22] for the binary branching random walk.…”

“…Proof. Again, by Lemma A.1 of [29], we can suppose that F ∈ C u b (D). Up to decomposing f into its positive and its negative part, we assume without loss of generality that f ≥ 0.…”

“…Formally, in the case β = ∞, the measure µ n,∞ is the uniform measure on the trajectories leading to the leftmost position at time n, which has been treated in [12]. For β < 1, Pain [29] proved that the trajectory of a particle chosen according to ν n,β behaves as a random walk with positive drift. In particular, after removing the drift, the rescaled trajectory converges toward a Brownian path.…”

Genealogy of the extremal process of the branching random walk

“…Chauvin and Rouault [CR97] studied similarly the overlap of subcritical measures, such that β < 1. They proved that in this case, the measure ω n,β converges toward δ 0 , and the measure λ n,β converges toward a proper probability measure on N. For the critical case, Pain [Pai17] proves that if (β n ) is a sequence converging to 1, then lim n→∞ ω n,βn = δ 0 in probability.…”

1-stable fluctuations in branching Brownian motion at critical temperature I: The derivative martingale

“…This method, which uses the probability Q x defined in the previous section, is due to Aïdékon [3] and sometimes called the peeling lemma, see Shi [75] for more applications. We also incorporate simplifications of this method from Pain [72]. Throughout, let x ≥ 0.…”