Derivative Martingale Research Articles

Critical Gaussian multiplicative chaos for singular measures

Given d≥1, we provide a construction of the random measure – the critical Gaussian Multiplicative Chaos – formally defined as e2dXdμ where X is a log-correlated Gaussian field and μ is a locally finite measure on Rd. Our construction generalizes the one performed in the case where μ is the Lebesgue measure. It requires that the measure μ is sufficiently spread out, namely that for μ almost every x we have ∫B(x,1)μ(dy)|x−y|deρlog1|x−y|<∞, where ρ:R+→R+ can be chosen to be any lower envelope function for the 3-Bessel process (this includes ρ(x)=xα with α∈(0,1/2)). We prove that three distinct random objects converge to a common limit which defines the critical GMC: the derivative martingale, the critical martingale, and the exponential of the mollified field. We also show that the above criterion for the measure μ is in a sense optimal.

1-stable fluctuation of the derivative martingale of branching random walk

In this paper, we study the functional convergence in law of the fluctuations of the derivative martingale of branching random walk on the real line. Our main result strengthens the results of Buraczewski et al. (2021) and is the branching random walk counterpart of the main result of Maillard and Pain (2019) for branching Brownian motion.

A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process

A continuous-time particle system on the real line satisfying the branching property and an exponential integrability condition is called a branching L\'evy process, and its law is characterized by a triplet $(\sigma^2,a,\Lambda)$. We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of $(\sigma^2,a,\Lambda)$. This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred L\'evy processes conditioned to stay positive.

Derivative martingale of the branching Brownian motion in dimension d≥1

On considère un mouvement brownien branchant dans Rd. Nous montrons qu’il existe presque sûrement un sous-ensemble aléatoire Θ de Sd−1 de mesure pleine tel que la limite de la martingale dérivée existe simultanément pour toutes les directions θ∈Θ. Cela nous permet de définir une mesure aléatoire sur Sd−1 dont la densité est donnée par la martingale dérivée. La preuve est basée sur des arguments de premier moment : nous approchons les martingales d’intérêt par une série de processus, qui ne prennent pas en compte les particules qui ont voyagé trop loin. Nous montrons que ces nouveaux processus sont des martingales uniformément intégrables dont les limites convergent vers les limites des martingales d’origine.

Critical Brownian multiplicative chaos

Brownian multiplicative chaos measures, introduced in Jego (Ann Probab 48:1597–1643, 2020), Aïdékon et al. (Ann Probab 48(4):1785–1825, 2020) and Bass et al. (Ann Probab 22:566–625, 1994), are random Borel measures that can be formally defined by exponentiating gamma times the square root of the local times of planar Brownian motion. So far, only the subcritical measures where the parameter gamma is less than 2 were studied. This article considers the critical case where gamma =2, using three different approximation procedures which all lead to the same universal measure. On the one hand, we exponentiate the square root of the local times of small circles and show convergence in the Seneta–Heyde normalisation as well as in the derivative martingale normalisation. On the other hand, we construct the critical measure as a limit of subcritical measures. This is the first example of a non-Gaussian critical multiplicative chaos. We are inspired by methods coming from critical Gaussian multiplicative chaos, but there are essential differences, the main one being the lack of Gaussianity which prevents the use of Kahane’s inequality and hence a priori controls. Instead, a continuity lemma is proved which makes it possible to use tools from stochastic calculus as an effective substitute.

On the derivative martingale in a branching random walk

We work under the Aïdékon–Chen conditions which ensure that the derivative martingale in a supercritical branching random walk on the line converges almost surely to a nondegenerate nonnegative random variable that we denote by Z. It is shown that EZ1{Z≤x}=logx+o(logx) as x→∞. Also, we provide necessary and sufficient conditions under which EZ1{Z≤x}=logx+const+o(1) as x→∞. This more precise asymptotics is a key tool for proving distributional limit theorems which quantify the rate of convergence of the derivative martingale to its limit Z. The methodological novelty of the present paper is a three terms representation of a subharmonic function of, at most, linear growth for a killed centered random walk of finite variance. This yields the aforementioned asymptotics and should also be applicable to other models.

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

Let $(Z_{t})_{t\geq 0}$ denote the derivative martingale of branching Brownian motion, that is, the derivative with respect to the inverse temperature of the normalized partition function at critical temperature. A well-known result by Lalley and Sellke (Ann. Probab. 15 (1987) 1052–1061) says that this martingale converges almost surely to a limit $Z_{\infty }$, positive on the event of survival. In this paper our concern is the fluctuations of the derivative martingale around its limit. A corollary of our results is the following convergence, confirming and strengthening a conjecture by Mueller and Munier (Phys. Rev. E 90 (2014) 042143): \begin{equation*}\sqrt{t}\bigg(Z_{\infty }-Z_{t}+\frac{\log t}{\sqrt{2\pi t}}Z_{\infty }\bigg)\xrightarrow[t\to \infty ]{}S_{Z_{\infty }}\quad\text{in law},\end{equation*} where $S$ is a spectrally positive 1-stable Lévy process independent of $Z_{\infty }$. In a first part of the paper, a relatively short proof of (a slightly stronger form of) this convergence is given based on the functional equation satisfied by the characteristic function of $Z_{\infty }$ together with tail asymptotics of this random variable. We then set up more elaborate arguments which yield a more thorough understanding of the trajectories of the particles contributing to the fluctuations. In this way we can upgrade our convergence result to functional convergence. This approach also sets the ground for a follow-up paper, where we study the fluctuations of more general functionals including the renormalized critical additive martingale. All proofs in this paper are given under the moment assumption $\mathbb{E}[L(\log L)^{3}]<\infty $, where the random variable $L$ follows the offspring distribution of the branching Brownian motion. We believe this hypothesis to be optimal.

Probability tilting of compensated fragmentations

Fragmentation processes are part of a broad class of models describing the evolution of a system of particles which split apart at random. These models are widely used in biology, materials science and nuclear physics, and their asymptotic behaviour at large times is interesting both mathematically and practically. The spine decomposition is a key tool in its study. In this work, we consider the class of compensated fragmentations, or homogeneous growth-fragmentations, recently defined by Bertoin. We give a complete spine decomposition of these processes in terms of a L\'evy process with immigration, and apply our result to study the asymptotic properties of the derivative martingale.

The near-critical Gibbs measure of the branching random walk

Considerons une marche aleatoire branchante surcritique reelle dans le cas frontiere et la mesure de Gibbs associee $\nu_{n,\beta}$ sur la $n$-ieme generation, qui est aussi la mesure de polymere sur un arbre desordonne avec temperature inverse $\beta$. La convergence de la fonction de partition $W_{n,\beta}$, apres renormalisation, vers une limite non-triviale a ete demontree par Aidekon et Shi (Ann. Probab. 42 (3) (2014) 959–993) dans le cas critique $\beta=1$ et par Madaule (J. Theoret. Probab. 30 (1) (2017) 27–63) pour $\beta>1$. On s’interesse ici au cas presque-critique, ou $\beta_{n}\to1$, et on montre la convergence de $W_{n,\beta_{n}}$, apres renormalisation, vers la limite de la martingale derivee a un facteur multiplicatif pres. D’autre part, les trajectoires de particules tirees selon la mesure de Gibbs $\nu_{n,\beta}$ ont ete etudiees par Madaule (Stochastic Process. Appl. 126 (2) (2016) 470–502) dans le cas critique, avec convergence vers le meandre brownien, et par Chen, Madaule et Mallein (On the trajectory of an individual chosen according to supercritical gibbs measure in the branching random walk (2015) Preprint) dans le regime de desordre fort, avec convergence vers l’excursion brownienne. On montre ici la convergence des trajectoires de particules tirees selon la mesure de Gibbs presque-critique et cela fait apparaitre une famille continue de processus allant du meandre jusqu’a l’excursion ou jusqu’au mouvement brownien.

On Seneta–Heyde scaling for a stable branching random walk

Abstract We consider a discrete-time branching random walk in the boundary case, where the associated random walk is in the domain of attraction of an α-stable law with 1 &lt; α &lt; 2. We prove that the derivative martingale Dn converges to a nontrivial limit D∞ under some regular conditions. We also study the additive martingale Wn and prove that n1/αWn converges in probability to a constant multiple of D∞.

High points of branching Brownian motion and McKean’s Martingale in the Bovier-Hartung extremal process

It has been proved by Bovier & Hartung [Elect. J. Probab. 19 (2014)] that the maximum of a variable-speed branching Brownian motion (BBM) in the weak correlation regime converges to a randomly shifted Gumbel distribution. The random shift is given by the almost sure limit of McKean’s martingale, and captures the early evolution of the system. In the Bovier-Hartung extremal process, McKean’s martingale thus plays a role which parallels that of the derivative martingale in the classical BBM. In this note, we provide an alternative interpretation of McKean’s martingale in terms of a law of large numbers for high-points of BBM, i.e. particles which lie at a macroscopic distance from the edge. At such scales, ‘McKean-like martingales’ are naturally expected to arise in all models belonging to the BBM-universality class.

Convergence of the centered maximum of log-correlated Gaussian fields

We show that the centered maximum of a sequence of log-correlated Gaussian fields in any dimension converges in distribution, under the assumption that the covariances of the fields converge in a suitable sense. We identify the limit as a randomly shifted Gumbel distribution, and characterize the random shift as the limit in distribution of a sequence of random variables, reminiscent of the derivative martingale in the theory of Branching Random Walk and Gaussian Chaos. We also discuss applications of the main convergence theorem and discuss examples that show that for logarithmically correlated fields, some additional structural assumptions of the type we make are needed for convergence of the centered maximum.

Extended convergence of the extremal process of branching Brownian motion

We extend the results of Arguin et al. [Probab. Theory Related Fields 157 (2013) 535–574] and Aïdékon et al. [Probab. Theory Related Fields 157 (2013) 405–451] on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the “location” of the particle in the underlying Galton–Watson tree. We show that the limit is a cluster point process on $\mathbb{R}_{+}\times\mathbb{R}$ where each cluster is the atom of a Poisson point process on $\mathbb{R}_{+}\times\mathbb{R}$ with a random intensity measure $Z(dz)\times C\mathrm{e}^{-\sqrt{2}x}\,dx$, where the random measure is explicitly constructed from the derivative martingale. This work is motivated by an analogous result for the Gaussian free field by Biskup and Louidor [Full extremal process, cluster law and freezing for two-dimensional discrete Gaussian free field (2016)].

Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field

We consider the discrete Gaussian Free Field in a square box in $\mathbb Z^2$ of side length $N$ with zero boundary conditions and study the joint law of its properly-centered extreme values ($h$) and their scaled spatial positions ($x$) in the limit as $N\to\infty$. Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an $r_N$-neighborhood thereof, we prove that the associated process tends, whenever $r_N\to\infty$ and $r_N/N\to0$, to a Poisson point process with intensity measure $Z(dx)e^{-\alpha h}dh$, where $\alpha:= 2/\sqrt{g}$ with $g:=2/\pi$ and where $Z(dx)$ is a random Borel measure on $[0,1]^2$. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure $Z$ is a version of the derivative martingale associated with the continuum Gaussian Free Field.

Maximum of a log-correlated Gaussian field

Nous étudions le maximum d’un champ Gaussien sur $[0,1]^{\mathtt{d}}$ ($\mathtt{d}\geq1$) dont les corrélations décroissent logarithmiquement avec la distance. Kahane (Ann. Sci. Math. Québec 9 (1985) 105–150) a introduit ce modèle pour construire mathématiquement le chaos Gaussien multiplicatif dans le cas sous-critique. Duplantier, Rhodes, Sheffield et Vargas (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint, Renormalization of critical Gaussian multiplicative chaos and KPZ formula (2012) Preprint) ont étendu cette construction au cas critique et ont établi la formule KPZ. De plus, dans (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint), ils fournissent plusieurs conjectures sur le cas sur-critique ainsi que sur le maximum de ce champ Gaussien. Dans ce papier nous établissons la convergence en loi du maximum et montrons que loi limite est une variable aléatoire de Gumbel convoluée avec la limite de la martingale dérivée, résolvant ainsi la Conjecture 12 de (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint).

A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk

We consider a branching random walk. Biggins and Kyprianou (2004) proved that, in the boundary case, the associated derivative martingale converges almost surely to a finite nonnegative limit, whose law serves as a fixed point of a smoothing transformation (Mandelbrot's cascade). In this paper, we give a necessary and sufficient condition for the nontriviality of the limit in this boundary case.

A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk

We consider a branching random walk. Biggins and Kyprianou (2004) proved that, in the boundary case, the associated derivative martingale converges almost surely to a finite nonnegative limit, whose law serves as a fixed point of a smoothing transformation (Mandelbrot's cascade). In this paper, we give a necessary and sufficient condition for the nontriviality of the limit in this boundary case.

The Glassy Phase of Complex Branching Brownian Motion

In this paper, we study complex valued branching Brownian motion in the so-called glassy phase, or also called phase II. In this context, we prove a limit theorem for the complex partition function hence confirming a conjecture formulated by Lacoin and the last two authors in a previous paper on complex Gaussian multiplicative chaos. We will show that the limiting partition function can be expressed as a product of a Gaussian random variable, mainly due to the windings of the phase, and a stable transform of the so called derivative martingale, mainly due to the clustering of the modulus. The proof relies on the fine description of the extremal process available in the branching Brownian motion context.

Fixed points of the multivariate smoothing transform: the critical case

Given a sequence T_1, T_2, ... of random d-by-d matrices with nonnegative entries, suppose there is a random vector X with nonnegative entries, such that the sum T_1 X_1 + T_2 X_2 +... has the same law as X, with X_1, X_2, ... being i.i.d.copies of X, independent of T_1, T_2, ... Then (the law of) X is called a fixed point of the multivariate smoothing transform. Similar to the well-studied one-dimensional case d=1, a function m is introduced, such that the existence of $\alpha \in (0,1]$ with $m(\alpha)=1$ and $m'(\alpha) \le 0$ guarantees the existence of nontrivial fixed points. We prove the uniqueness of fixed points in the critical case $m'(\alpha)=0$ and describe their tail behavior. This complements recent results for the non-critical multivariate case. Moreover, we introduce the multivariate analogue of the derivative martingale and prove its convergence to a non-trivial limit.

On Barnes beta distributions, Selberg integral and Riemann xi

Abstract The theory of Barnes beta probability distributions is advanced and related to the Riemann xi function. The scaling invariance, multiplication formula, and Shintani factorization of Barnes multiple gamma functions are reviewed using the approach of Ruijsenaars and shown to imply novel properties of Barnes beta distributions. The applications are given to the meromorphic extension of the Selberg integral as a function of its dimension and the scaling invariance of the underlying probability distribution. This probability distribution in the critical case is described and conjectured to be the distribution of the derivative martingale. The Jacobi triple product is interpreted probabilistically resulting in an approximation of the Riemann xi function by the Mellin transform of the logarithm of a limit of Barnes beta distributions.