Description:    It has been conjectured since the work of Lalley and Sellke (Ann. Probab., 15, 1052–1061, 1987 ) that branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. Very recently, it emerged that this can be proved in several different ways (see e.g. Brunet and Derrida, A branching random walk seen from the tip, 2010 , Poissonian statistics in the extremal process of branching Brownian motion, 2010 ; Arguin et al., The extremal process of branching Brownian motion, 2011 ). The structure of this extremal point process turns out to be a Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process. The main goal of the present work is to give a complete description of the limit object via an explicit construction of this decoration point process. Another proof and description has been obtained independently by Arguin et al. (The extremal process of branching Brownian motion, 2011 ). Content Type Journal Article Pages 1-47 DOI 10.1007/s00440-012-0461-0 Authors E. Aïdékon, Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, Eindhoven, The Netherlands J. Berestycki, Laboratoire de Probabilités et Modèles Aléatoires, CNRS UMR 7599, UPMC Université Paris 6, Case courrier 188, 4, Place Jussieu, 75252  Paris Cedex 05, France É. Brunet, Laboratoire de Physique Statistique, École Normale Supérieure, UPMC Université Paris 6, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005  Paris, France Z. Shi, Laboratoire de Probabilités et Modèles Aléatoires, CNRS UMR 7599, UPMC Université Paris 6, Case courrier 188, 4, Place Jussieu, 75252  Paris Cedex 05, France Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051

Description:    It is shown that the normalized fluctuations of Riemann’s zeta zeros around their predicted locations follow the Gaussian law. It is also shown that fluctuations of two zeros, g k and g k + x , with $$x\sim \left( \log k\right) ^{\beta }, \beta 〉0,$$ for large k follow the two-variate Gaussian distribution with correlation ( 1 - b ) + . Content Type Journal Article Pages 1-30 DOI 10.1007/s00440-012-0465-9 Authors V. Kargin, Statistical Laboratory, Department of Mathematics, University of Cambridge, Cambridge, UK Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051

Description:    We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the clusters is a Poisson process with intensity measure with exponential density. The law of the individual clusters is characterized as branching Brownian motions conditioned to perform “unusually large displacements”, and its existence is proved. The proof combines three main ingredients. First, the results of Bramson on the convergence of solutions of the Kolmogorov–Petrovsky–Piscounov equation with general initial conditions to standing waves. Second, the integral representations of such waves as first obtained by Lalley and Sellke in the case of Heaviside initial conditions. Third, a proper identification of the tail of the extremal process with an auxiliary process (based on the work of Chauvin and Rouault), which fully captures the large time asymptotics of the extremal process. The analysis through the auxiliary process is a rigorous formulation of the cavity method developed in the study of mean field spin glasses. Content Type Journal Article Pages 1-40 DOI 10.1007/s00440-012-0464-x Authors Louis-Pierre Arguin, Département de Mathématiques et de Statistique, Université de Montréal, 2920 Chemin de la Tour,  Montréal,  H3T 1J4 Canada Anton Bovier, Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany Nicola Kistler, Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051

Description:    Motivated by earlier work on the use of fully-coupled forward–backward stochastic differential equations (henceforth FBSDEs) in the analysis of mathematical models for the CO 2 emissions markets, the present study is concerned with the analysis of these equations when the generator of the forward equation has a conservative degenerate structure and the terminal condition of the backward equation is a non-smooth function of the terminal value of the forward component. We show that a general form of existence and uniqueness result still holds. When the function giving the terminal condition is binary, we also show that the flow property of the forward component of the solution can fail at the terminal time. In particular, we prove that a Dirac point mass appears in its distribution, exactly at the location of the jump of the binary function giving the terminal condition. We provide a detailed analysis of the breakdown of the Markovian representation of the solution at the terminal time. Content Type Journal Article Pages 1-56 DOI 10.1007/s00440-012-0459-7 Authors René Carmona, ORFE, Bendheim Center for Finance, Princeton University, Princeton, NJ 08544, USA François Delarue, Laboratoire J.-A. Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Cedex 02, Nice, France Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051

Description:    We consider the problems of chaos in disorder and temperature for coupled copies of the mixed p -spin models. Under certain assumptions on the parameters of the models, first we prove a weak form of chaos by showing that the overlap is concentrated around its Gibbs average depending on the disorder and then obtain several results toward strong chaos by providing control of the overlap between two systems in terms of their Parisi measures. Content Type Journal Article Pages 1-16 DOI 10.1007/s00440-012-0460-1 Authors Wei-Kuo Chen, Department of Mathematics, University of California, Irvine, USA Dmitry Panchenko, Department of Mathematics, Texas A&M University, College Station, USA Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051

Description:    In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions ( d ³ 5 ). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon as a non-trivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cut-times to relate the associated biased random walk to a one-dimensional random walk in a random environment in Sinai’s regime. Via this approach, a corresponding aging result is also proved. Content Type Journal Article Pages 1-20 DOI 10.1007/s00440-012-0463-y Authors David A. Croydon, Department of Statistics, University of Warwick, Coventry, CV4 7AL UK Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051

Description:    We consider the biased random walk on a critical Galton–Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain one-dimensional trapping models with slowly-varying tails. Indeed, in each of these two settings, we establish closely-related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs. Content Type Journal Article Pages 1-55 DOI 10.1007/s00440-012-0462-z Authors D. A. Croydon, Department of Statistics, University of Warwick, Coventry, CV4 7AL UK A. Fribergh, CIMS, New York University, 251 Mercer Street, New York, 10012-1185 USA T. Kumagai, RIMS, Kyoto University, Kyoto, 606-8502 Japan Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051

Description:    We study the thermal properties of a pinned disordered harmonic chain weakly perturbed by a noise and an anharmonic potential. The noise is controlled by a parameter l ® 0 , and the anharmonicity by a parameter l ¢ £ l . Let k be the conductivity of the chain, defined through the Green–Kubo formula. Under suitable hypotheses, we show that k = O ( l ) and, in the absence of anharmonic potential, that k ~ l . This is in sharp contrast with the ordered chain for which k ~ 1/ l , and so shows the persistence of localization effects for a non-integrable dynamics. Content Type Journal Article Pages 1-31 DOI 10.1007/s00440-012-0458-8 Authors Cédric Bernardin, Université de Lyon and CNRS, UMPA, UMR-CNRS 5669, ENS-Lyon, 46, allée d’Italie, 69364 Lyon Cedex 07, France François Huveneers, CEREMADE, Université de Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051

Description:    This paper extends recent results on ageing in mean field spin glasses on short time scales, obtained by Ben Arous and Gün (Commun Pure Appl Math 65:77–127, 2012 ) in law with respect to the environment, to results that hold almost surely, respectively in probability, with respect to the environment. It is based on the methods put forward in (Gayrard in Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM, 2010 ; Electron J Probab 17(58): 1–33, 2012 ) and naturally complements (Bovier and Gayrard in Ann Probab, 2012 ). Content Type Journal Article Pages 1-33 DOI 10.1007/s00440-012-0456-x Authors Anton Bovier, Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität, Endenicher Allee 60, 53115 Bonn, Germany Véronique Gayrard, CMI, LAPT, Université de Provence, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13, France Adéla Švejda, Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität, Endenicher Allee 60, 53115 Bonn, Germany Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051

Description:    We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein–Orlicz norm. This new norm interpolates sub-Gaussian and sub-exponential tail behavior. In particular, we show how this norm can be used to simplify the derivation of deviation inequalities for suprema of collections of random variables. Secondly, we introduce chaining and generic chaining along a tree. These simplify the well-known concepts of chaining and generic chaining. The supremum of the empirical process is then studied as a special case. We show that chaining along a tree can be done using entropy with bracketing. Finally, we establish a deviation inequality for the empirical process for the unbounded case. Content Type Journal Article Pages 1-26 DOI 10.1007/s00440-012-0455-y Authors Sara van de Geer, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland Johannes Lederer, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland Journal Probability Theory and Related Fields Online ISSN 1432-2064 Print ISSN 0178-8051