The branching capacity has been introduced by [Zhu 2016] as the limit of the hitting probability of a symmetric branching random walk in $\mathbb Z^d$, $d\ge 5$. Similarly, we define the Brownian snake capacity in $\mathbb R^d$, as the scaling limit of the hitting probability by the Brownian snake starting from afar. Then, we prove our main result on the vague convergence of the rescaled branching capacity towards this Brownian snake capacity. Our proof relies on a precise convergence rate for the approximation of the branching capacity by hitting probabilities.

We study the configurations of the nearest neighbor Ising ferromagnetic chain with IID centered and square integrable external random field in the limit in which the pairwise interaction tends to infinity. The available free energy estimates for this model show a strong form of disorder relevance, i.e., a strong effect of disorder on the free energy behavior, and our aim is to make explicit how the disorder affects the spin configurations. We give a quantitative estimate that shows that the infinite volume spin configurations are close to one explicit disorder dependent configuration when the interaction is large. Our results confirm predictions on this model obtained in D. S. Fisher, P. Le Doussal and C. Monthus (Phys. Rev. E 2001) by applying the renormalization group method introduced by D. S. Fisher (Phys. Rev. B 1995).

The problem of conditioning on the occupation field was investigated for the Brownian motion in 1998 independently by Aldous [4] and Warren and Yor [34] and recently for the loop soup at intensity $1/2$ by Werner [35], Sabot and Tarr\`es [30], and Lupu, Sabot and Tarr\`es [22]. We consider this problem in the case of the Brownian loop soup on the real line, and show that it is connected with a flow version of Jacobi processes, called Jacobi flow. We give a pathwise construction of this flow simultaneously for all parameters by means of a common Brownian motion, via the perturbed reflecting Brownian motion. The Jacobi flow is related to Fleming--Viot processes, as established by Bertoin and Le Gall [9] and Dawson and Li [11]. This relation allows us to interpret Perkins' disintegration theorem between Feller continuous state branching-processes and Fleming--Viot processes as a decomposition of Gaussian measures. Our approach gives a unified framework for the problems of disintegrating on the real line. The connection with Bass--Burdzy flows which was drawn in Warren [33] and Lupu, Sabot and Tarr\`es [23] is shown to be valid in the general case.

Let (V(u), u∈T) be a (supercritical) branching random walk and (η_u, u∈T) be marks on the vertices of the tree, distributed in an i.i.d. fashion. Following Aldous and Bandyopadhyay [Ann. Appl. Probab., 15, 1047–1110 (2005)], for each ray ξ of the tree, we associate the discounted tree sum D(ξ) which is the sum of the e^-V(u)η_u taken along the ray. The paper deals with the finiteness of supξD(ξ). To this end, we study the extreme behaviour of the local time processes of the paths (V(u), u ∈ ξ). It answers a question of Nicolas Curien, and partially solves Open Problem 31 of Aldous and Bandyopadhyay [Ann. Appl. Probab., 15, 1047–1110 (2005)]. We also present several open questions.

We are interested in the branching capacity of the range of a random walk in Zd. Schapira [29] has recently obtained precise asymptotics in the case d >= 6 and has demonstrated a transition at dimension d = 6. We study the case d = 5 and prove that the renormalized branching capacity converges in law to the Brownian snake capacity of the range of a Brownian motion. The main step in the proof relies on studying the intersection probability between the range of a critical Branching random walk and that of a random walk, which is of independent interest.

Attach to each edge of the complete graph on n vertices, i.i.d. exponential random variables with mean n. Aldous [1] proved that the longest path with average weight below p undergoes a phase transition at p=1/e: it is o(n) when p<1/e and of order n if p>1/e. Later, Ding [4] revealed a finer phase transition around 1/e: there exist c'>c>0 such that the length of the longest path is of order ln^3 n if p ≤1/e+c/ln^2 n and is polynomial if p≥1/e+c'/ln^2 n. We identify the location of this phase transition and obtain sharp asymptotics of the length near criticality. The proof uses an exploration mechanism mimicking a branching random walk with selection introduced by Brunet and Derrida [3].

We consider a generalized Derrida-Retaux model on a Galton-Watson tree with a geometric offspring distribution. For a class of recursive systems, including the Derrida-Retaux model with either a geometric or exponential initial distribution, we characterize the critical curve using an involution-type equation and prove that the free energy satisfies the Derrida-Retaux conjecture.

We study the top Lyapunov exponent of a product of random 2 × 2 matrices appearing in the analysis of several statistical mechanical models with disorder, extending a previous treatment of the critical case (Giacomin and Greenblatt, ALEA 19 (2022), 701-728) by significantly weakening the assumptions on the disorder distribution. The argument we give completely revisits and improves the previous proof. As a key novelty we build a probability that is close to the Furstenberg probability, i.e. the invariant probability of the Markov chain corresponding to the evolution of the direction of a vector in ℝ^2 under the action of the random matrices, in terms of the ladder times of a centered random walk which is directly related to the random matrix sequence. We then show that sharp estimates on the ladder times (renewal) process lead to a sharp control on the probability measure we build and, in turn, to the control of its distance from the Furstenberg probability.

The biased random walk on supercritical Galton–Watson trees is known to exhibit a multiscale phenomenon in the slow regime: the maximal displacement of the walk in the first n steps is of order (log n)3, whereas the typical displacement of the walk at the n-th step is of order (log n)2. Our main result reveals another multiscale property of biased walks: the maximal potential energy of the biased walks is of order (log n)2 in contrast with its typical size, which is of order log n. The proof relies on analyzing the intricate multiscale structure of the potential energy.

In this paper, we study second order fluctuations for the size of the range of a critical branching random walk (BRW) in ℤ^d. We consider the BRW with geometric offspring indexed by the Kesten tree, and show that the size of its range has linear variance when d>8, and satisfies a central limit theorem (CLT) with Gaussian limiting distribution when d>16. The proof relies on the stationarity of the model under depth-first exploration, a general CLT by Dedecker and Merlevède [7], a truncation technique exploiting the local independence of tree structures, and a recursion argument for moment bounds.

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 $(p-p_c)^{-\frac12}$ as $p \downarrow p_c$. We study the nearly subcritical regime ($p \uparrow p_c$) and aim at a dual version of the Derrida-Retaux conjecture; our main result states that as $n \to \infty$, both $\E(X_n)$ and $\P(X_n\neq 0)$ decay exponentially with exponent $(p_c-p)^{\frac12 +o(1)}$, where $o(1) \to 0$ as $p \uparrow p_c$.

We consider the continuum version of the random field Ising model in one dimension: this model arises naturally as weak disorder scaling limit of the original Ising model. Like for the Ising model, a spin configuration is conveniently described as a sequence of spin domains with alternating signs (domain-wall structure). We show that for fixed centered external field and as spin-spin couplings become large, the domain-wall structure scales to a disorder dependent limit that coincides with the infinite disorder fixed point process introduced by D. S. Fisher in the context of zero temperature quantum Ising chains. In particular, our results establish a number of predictions that one can find in Fisher et al. (Phys Rev E 64:41, 2001). The infinite disorder fixed point process for centered external field is equivalently described in terms of the process of suitably selected extrema of a Brownian trajectory introduced and studied by Neveu and Pitman (in: Séminaire de probabilités XXIII. Lecture notes in mathematics, vol 1372, pp 239–247, 1989). This characterization of the infinite disorder fixed point is one of the important ingredients of our analysis.

The classical Ray-Knight theorems for the Brownian motion determine the law of its local time process either at the first hitting time of a given value a by the local time at the origin, or at the first hitting time of a given position b by the Brownian motion. We extend these results by describing the local time process jointly for all a and b , by means of the stochastic integral with respect to an appropriate white noise. Our result applies to μ -processes, and has an immediate application: a μ -process is the height process of a Feller continuous-state branching process (CSBP) with immigration (Lambert (2002)), whereas a Feller CSBP with immigration satisfies a stochastic differential equation (SDE) driven by a white noise (Dawson and Li (2012)); our result gives an explicit relation between these two descriptions and shows that the SDE in question is a reformulation of Tanaka’s formula.

Let $R_n$ be the range of a critical branching random walk with $n$ particles on $\mathbb Z^d$, which is the set of sites visited by a random walk indexed by a critical Galton--Watson tree conditioned on having exactly $n$ vertices. For $d\in\{3, 4, 5\}$, we prove that $n^{-\frac{d-2}4} \mathtt{cap}^{(d)}(R_n)$, the renormalized capacity of $R_n$, converges in law to the capacity of the support of the integrated super-Brownian excursion. The proof relies on a study of the intersection probabilities between the critical branching random walk and an independent simple random walk on $\mathbb Z^d$.

Consider a branching random walk $$(V_u)_{u\in\mathcal T^{\text{IGW}}}$$ in $$\mathbb Z^d$$ with the genealogy tree $$\mathcal T^{\text{IGW}}$$ formed by a sequence of i.i.d. critical Galton–Watson trees. Let $$R_n$$ be the set of points in $$\mathbb Z^d$$ visited by $$(V_u)$$ when the index $$u$$ explores the first $$n$$ subtrees in $$\mathcal T^{\text{IGW}}$$ . Our main result states that for $$d\in\{3,4,5\}$$ , the capacity of $$R_n$$ is almost surely equal to $$n^{(d-2)/{2}+o(1)}$$ as $$n\to\infty$$ .

Рассматривается ветвящееся случайное блуждание $(V_u)_{u\in \mathcal T^{\mathrm{IGW}}}$ в пространстве $\mathbb Z^d$ с генеалогическим деревом $\mathcal T^{\mathrm{IGW}}$, образованным последовательностью независимых одинаково распределенных критических деревьев Гальтона-Ватсона. Пусть $R_n$ - множество точек в пространстве $\mathbb Z^d$, которые посещает ветвящееся случайное блуждание $(V_u)$, когда индекс $u$ пробегает первые $n$ поддеревьев дерева $\mathcal T^{\mathrm{IGW}}$. Основной результат работы состоит в том, что для $d\in \{3,4,5\}$ при $n\to \infty $ емкость множества $R_n$ имеет почти наверное порядок $n^{(d-2)/{2}+o(1)}$.

We are interested in the recursive model $$(Y_n, \, n\ge 0)$$ studied by Collet et al. (Commun Math Phys 94:353–370, 1984) and by Derrida and Retaux (J Stat Phys 156:268–290, 2014). We prove that at criticality, the probability $$\mathbf{P}(Y_n>0)$$ behaves like $$n^{-2 + o(1)}$$ as n goes to infinity; this gives a weaker confirmation of predictions made in Collet et al. (1984), Derrida and Retaux (2014) and Chen et al. (in: Sidoravicius (ed) Sojourns in probability theory and statistical physics-III, Springer, Singapore, 2019). Our method relies on studying the number of pivotal vertices and open paths, combined with a delicate coupling argument.

We are interested in path decompositions of a perturbed reflecting Brownian motion (PRBM) at the hitting times and at the minimum. Our study relies on the loop soups developed by Lawler and Werner [10] and Le Jan [13]-[14], in particular on a result discovered by Lupu [15] identifying the law of the excursions of the PRBM above its past minimum with the loop measure of Brownian bridges.

We are interested in the nearly supercritical regime in a family of max-type recursive models studied by Collet, Eckman, Glaser and Martin and by Derrida and Retaux, and prove that under a suitable integrability assumption on the initial distribution, the free energy vanishes at the transition with an essential singularity with exponent $\tfrac12$. This gives a weaker answer to a conjecture of Derrida and Retaux. Other behaviours are obtained when the integrability condition is not satisfied.

To study the depinning transition in the limit of strong disorder, Derrida and Retaux (J Stat Phys 156(2):26–290, 2014) 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.