Search

Your search keyword '"Bruno Schapira"' showing total 25 results
Start Over Author "Bruno Schapira" Topic statistics and probability
25 results on '"Bruno Schapira"'

1. Once reinforced random walk on Z×Γ

Author

Bruno Schapira, Daniel Kious, and Arvind Singh

Subjects
Statistics and Probability, 010104 statistics & probability, 010102 general mathematics, 0101 mathematics, Statistics, Probability and Uncertainty, Random walk, 01 natural sciences, Humanities, Mathematics
Abstract

Nous revisitons un article de Vervoort (Vervoort (2002)), jamais publie, sur la marche une-fois-renforcee, ou once reinforced random walk, et nous prouvons que ce processus est recurrent sur tout graphe de la forme Z×Γ, ou Γ est un graphe fini, et pour un parametre de renforcement suffisamment grand. Nous obtenons egalement un theoreme de forme pour l’ensemble des sites visites, et prouvons que les fluctuations sont d’ordre polynomial. La preuve utilise des estimees precises et generales sur le temps passe dans des sous-graphes du graphe ambiant, ce qui pourrait etre interessant par soi-meme.

Published
2021
Full Text
View/download PDF

2. The two regimes of moderate deviations for the range of a transient walk

Author

Bruno Schapira, Amine Asselah, Université Paris-Est Créteil Val-de-Marne - Faculté des sciences et technologie (UPEC FST), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2016), and ANR-17-CE40-0032,SWiWS,Gruyère et Saucisse de Wiener(2017)

Subjects
Statistics and Probability, Mathematical finance, Gaussian, 010102 general mathematics, Range, Random walk, 01 natural sciences, Upper and lower bounds, Exponential function, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, symbols.namesake, Large deviations, Moderate deviations, Dimension (vector space), Norm (mathematics), Random Walk, Range (statistics), symbols, Capacity, Statistical physics, 0101 mathematics, Statistics, Probability and Uncertainty, Analysis, Mathematics
Abstract

We obtain sharp upper and lower bounds for the downward moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched exponential regime for larger deviations. In the latter regime, we show that conditioned on the moderate deviations event, the walk folds a small part of its range in a ball-like subset. Also, we provide new path properties, in dimension three as well. Besides the key role Newtonian capacity plays in this study, we introduce two original ideas, of general interest, which strengthen the approach developed in Asselah and Schapira (Sci Ec Norm Super 50(4):755–786, 2017).

Published
2021
Full Text
View/download PDF

3. Capacity of the range in dimension $5$

Author

Bruno Schapira

Subjects
Statistics and Probability, Discrete mathematics, 60G50, Logarithm, capacity, central limit theorem, intersection of random walk ranges, Random walk, Moment (mathematics), Distribution (mathematics), Intersection, Dimension (vector space), 60F05, range, 60J45, Range (statistics), Statistics, Probability and Uncertainty, Central limit theorem, Mathematics
Abstract

We prove a central limit theorem for the capacity of the range of a symmetric random walk on $\mathbb{Z}^{5}$, under only a moment condition on the step distribution. The result is analogous to the central limit theorem for the size of the range in dimension three, obtained by Jain and Pruitt in 1971. In particular, an atypical logarithmic correction appears in the scaling of the variance. The proof is based on new asymptotic estimates, which hold in any dimension $d\ge5$, for the probability that the ranges of two independent random walks intersect. The latter are then used for computing covariances of some intersection events at the leading order.

Published
2020
Full Text
View/download PDF

4. The contact process on random hyperbolic graphs: metastability and critical exponents

Author

Dieter Mitsche, Daniel Valesin, Amitai Linker, Bruno Schapira, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), University of Groningen [Groningen], Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Schapira, Bruno, and Stochastic Studies and Statistics

Subjects
Statistics and Probability, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Second moment of area, 01 natural sciences, Power law, 010104 statistics & probability, Metastability, FOS: Mathematics, 82C22, 05C80, Mathematics - Combinatorics, 0101 mathematics, Mathematics, Random graph, 010102 general mathematics, Mathematical analysis, random hyperbolic graphs, Probability (math.PR), Degree distribution, Universality (dynamical systems), Contact process, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Exponent, Combinatorics (math.CO), Statistics, Probability and Uncertainty, Critical exponent, Mathematics - Probability
Abstract

International audience; We consider the contact process on the model of hyperbolic random graph, in the regime when the degree distribution obeys a power law with exponent χ ∈ (1, 2) (so that the degree distribution has finite mean and infinite second moment). We show that the probability of non-extinction as the rate of infection goes to zero decays as a power law with an exponent that only depends on χ and which is the same as in the configuration model, suggesting some universality of this critical exponent. We also consider finite versions of the hyperbolic graph and prove metastability results, as the size of the graph goes to infinity.

Published
2020
Full Text
View/download PDF

5. On the nature of the Swiss cheese in dimension $3$

Author

Bruno Schapira and Amine Asselah

Subjects
Statistics and Probability, Discrete mathematics, 60G50, Optimal density, Random walk, large deviations, Mathematics::Probability, Dimension (vector space), range, Small range, Swiss cheese, Range (statistics), Large deviations theory, Statistics, Probability and Uncertainty, Rate function, 60F10, Mathematics
Abstract

We study scenarii linked with the Swiss cheese picture in dimension 3 obtained when two random walks are forced to meet often, or when one random walk is forced to squeeze its range. In the case of two random walks, we show that they most likely meet in a region of optimal density. In the case of one random walk, we show that a small range is reached by a strategy uniform in time. Both results rely on an original inequality estimating the cost of visiting sparse sites, and in the case of one random walk on the precise large deviation principle of van den Berg, Bolthausen and den Hollander (Ann. of Math. (2) 153 (2001) 355–406), including their sharp estimates of the rate functions in the neighborhood of the origin.

Published
2020
Full Text
View/download PDF

6. LOCALIZATION ON 5 SITES FOR VERTEX REINFORCED RANDOM WALKS: TOWARDS A CHARACTERIZATION

Author

Bruno Schapira, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2017), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), and ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2016)

Subjects
Statistics and Probability, Vertex (graph theory), Conjecture, Self-interacting random walk, 010102 general mathematics, Probability (math.PR), Integer lattice, Reinforced Random Walk, Characterization (mathematics), Random walk, 01 natural sciences, Combinatorics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010104 statistics & probability, Mathematics::Probability, FOS: Mathematics, 0101 mathematics, Statistics, Probability and Uncertainty, Positive probability, Mathematics - Probability, Mathematics
Abstract

We continue the investigation of the localization phenomenon for a Vertex Reinforced Random Walk on the integer lattice. We provide some partial results towards a full characterization of the weights for which localization on 5 sites occurs with positive probability, and make some conjecture concerning the almost sure behavior., Comment: 14 pages, Revised version. Accepted to Ann. Appl. Probab

Published
2019

7. Strong law of large numbers for the capacity of the Wiener sausage in dimension four

Author

Perla Sousi, Amine Asselah, Bruno Schapira, Apollo - University of Cambridge Repository, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2016), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), DPMMS/CMS, University of Cambridge [UK] (CAM), ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2017), Schapira, Bruno, and Marches aléatoires en interaction - - MALIN2016 - ANR-16-CE93-0003 - AAPG2016 - VALID

Subjects
Statistics and Probability, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Logarithm, Law of large numbers, Wiener sausage, Expected value, 01 natural sciences, and phrases Capacity, 010104 statistics & probability, symbols.namesake, Dimension (vector space), Intersection, Mathematics::Probability, FOS: Mathematics, Law of large, 0101 mathematics, Scaling, Mathematics, Capacity, 010102 general mathematics, Mathematical analysis, Probability (math.PR), 4901 Applied Mathematics, 16. Peace & justice, Green kernel, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 4905 Statistics, symbols, 49 Mathematical Sciences, Statistics, Probability and Uncertainty, Critical dimension, Analysis, Mathematics - Probability, 60F05, 60G50
Abstract

We prove a strong law of large numbers for the Newtonian capacity of a Wiener sausage in the critical dimension four., Comment: Substantially expanded: the asymptotic of the first moment is obtained, giving rise to a fully explicit law of large numbers

Published
2019

8. Deviations for the Capacity of the Range of a Random Walk

Author

Bruno Schapira, Amine Asselah, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2017), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2016), ANR-17-CE40-0032,SWiWS,Gruyère et Saucisse de Wiener(2017), Schapira, Bruno, Marches aléatoires en interaction - - MALIN2016 - ANR-16-CE93-0003 - AAPG2016 - VALID, Gruyère et Saucisse de Wiener - - SWiWS2017 - ANR-17-CE40-0032 - AAPG2017 - VALID, and Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU)

Subjects
Statistics and Probability, 60G50, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], 01 natural sciences, 010104 statistics & probability, Moderate deviations, Mathematics::Probability, MSC 2010 subject classifications. Primary 60F05, 60G50, 60F05, FOS: Mathematics, 0101 mathematics, Mathematics, Capacity, Probability (math.PR), 010102 general mathematics, Mathematical analysis, Range, Random walk, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Large deviations, Random Walk, Large deviations theory, Statistics, Probability and Uncertainty, Martingale (probability theory), Rate function, Mathematics - Probability
Abstract

We obtain estimates for large and moderate deviations for the capacity of the range of a random walk on $\mathbb{Z}^d$, in dimension $d\ge 5$, both in the upward and downward directions. The results are analogous to those we obtained for the volume of the range in two companion papers [AS17, AS19]. Interestingly, the main steps of the strategy we developed for the latter apply in this seemingly different setting, yet the details of the analysis are different, Comment: This version covers the whole moderate deviation regime in $d\ge 7$

Published
2018

9. Extinction time for the contact process on general graphs

Author

Daniel Valesin, Bruno Schapira, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), University of Groningen [Groningen], Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), and Dynamical Systems, Geometry & Mathematical Physics

Subjects
Statistics and Probability, Exponential distribution, PERCOLATION, contact process, 82C22, 60K35, Lambda, 01 natural sciences, extinction time, Combinatorics, 010104 statistics & probability, Exponential growth, FOS: Mathematics, 0101 mathematics, Finite set, Mathematics, Contact process, FINITE-SET, Mathematical finance, 010102 general mathematics, Probability (math.PR), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Extinction time, METASTABILITY, Bounded function, Statistics, Probability and Uncertainty, Analysis, Mathematics - Probability
Abstract

We consider the contact process on finite and connected graphs and study the behavior of the extinction time, that is, the amount of time that it takes for the infection to disappear in the process started from full occupancy. We prove, without any restriction on the graph $G$, that if the infection rate $\lambda$ is larger than the critical rate of the one-dimensional process, then the extinction time grows faster than $\exp\{|G|/(\log|G|)^\kappa\}$ for any constant $\kappa > 1$, where $|G|$ denotes the number of vertices of $G$. Also for general graphs, we show that the extinction time divided by its expectation converges in distribution, as the number of vertices tends to infinity, to the exponential distribution with parameter 1. These results complement earlier work of Mountford, Mourrat, Valesin and Yao, in which only graphs of bounded degrees were considered, and the extinction time was shown to grow exponentially in $n$; here we also provide a simpler proof of this fact., Comment: 23 pages

Published
2017
Full Text
View/download PDF

10. Boundary of the Range of Transient Random Walk

Author

Bruno Schapira, Amine Asselah, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), ANR-11-IDEX-0001,Amidex,INITIATIVE D'EXCELLENCE AIX MARSEILLE UNIVERSITE(2011), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), and ANR-11-IDEX-0001-02/11-IDEX-0001,AMIDEX,AMIDEX(2011)

Subjects
Statistics and Probability, Probability (math.PR), 010102 general mathematics, central limit theorem, 60K35, 82C22, 60J25, 60K35, 82C22, 60J25, Binary logarithm, Simple random sample, Random walk, 01 natural sciences, Upper and lower bounds, Combinatorics, random walk, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], boundary range, 0103 physical sciences, FOS: Mathematics, 010307 mathematical physics, 0101 mathematics, Statistics, Probability and Uncertainty, Martingale (probability theory), Mathematics - Probability, Analysis, Central limit theorem, Mathematics
Abstract

We study the boundary of the range of simple random walk on $\mathbb{Z}^d$ in the transient regime $d\ge 3$. We show that volumes of the range and its boundary differ mainly by a martingale. As a consequence, we obtain a bound on the variance of order $n\log n$ in dimension three. We also establish a central limit theorem in dimension four and larger., Comment: 23 pages. Revised Version

Published
2017
Full Text
View/download PDF

11. Localization of a vertex reinforced random walk on $$\mathbb{Z }$$ with sub-linear weight

Author

Arvind Singh, Anne-Laure Basdevant, and Bruno Schapira

Subjects
Statistics and Probability, Vertex (graph theory), Discrete mathematics, Weight function, Mathematical finance, Integer lattice, Interval (mathematics), Random walk, Set (abstract data type), Combinatorics, Statistics, Probability and Uncertainty, Analysis, Positive probability, Mathematics
Abstract

We consider a vertex reinforced random walk on the integer lattice with sub-linear reinforcement. Under some assumptions on the regular variation of the weight function, we characterize whether the walk gets stuck on a finite interval. When this happens, we estimate the size of the localization set. In particular, we show that, for any odd number $$N$$ larger than or equal to $$5$$ , there exists a vertex reinforced random walk which localizes with positive probability on exactly $$N$$ consecutive sites.

Published
2013
Full Text
View/download PDF

12. Capacity of the range of random walk on $\mathbb Z^4$

Author

Bruno Schapira, Perla Sousi, Amine Asselah, Schapira, Bruno, Marches aléatoires en interaction - - MALIN2016 - ANR-16-CE93-0003 - AAPG2016 - VALID, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Marseille (I2M), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), DPMMS/CMS, University of Cambridge [UK] (CAM), ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2017), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM), and ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2016)

Subjects
Statistics and Probability, 60G50, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], Central limit theorem, Integer lattice, Law of large numbers, 01 natural sciences, 010104 statistics & probability, Mathematics::Probability, Dimension (vector space), 60F05, FOS: Mathematics, 60F50, Limit (mathematics), 0101 mathematics, Mathematics, Discrete mathematics, Capacity, Probability (math.PR), 010102 general mathematics, Random walk, Green kernel, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Range (mathematics), Scaling limit, Statistics, Probability and Uncertainty, Mathematics - Probability
Abstract

We study the scaling limit of the capacity of the range of a random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-Gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in ’86 [Comm. Math. Phys. 104 (1986) 471–507] for the volume of the range in dimension two.

Published
2016

13. The Heckman–Opdam Markov processes

Author

Bruno Schapira

Subjects
Statistics and Probability, Pure mathematics, Mathematical finance, Mathematical analysis, Markov process, symbols.namesake, Semimartingale, Probability theory, Law of large numbers, Symmetric space, symbols, Decomposition method (constraint satisfaction), Statistics, Probability and Uncertainty, Analysis, Central limit theorem, Mathematics
Abstract

We introduce and study the natural counterpart of the Dunkl Markov processes in a negatively curved setting. We give a semimartingale decomposition of the radial part, and some properties of the jumps. We prove also a law of large numbers, a central limit theorem, and the convergence of the normalized process to the Dunkl process. Eventually we describe the asymptotic behavior of the infinite loop as it was done by Anker, Bougerol and Jeulin in the symmetric spaces setting in [1].

Published
2006
Full Text
View/download PDF

14. Metastability for the contact process on the configuration model with infinite mean degree

Author

Van Hao Can, Bruno Schapira, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), and ANR-10-BLAN-0125,MEMEMO 2,Random walks, Random Environments, Reinforcement.(2010)

Subjects
Statistics and Probability, Exponential distribution, 05C80, metastability, Exponential growth, Metastability, FOS: Mathematics, random graphs, Mathematics, Random graph, Discrete mathematics, Degree (graph theory), Mathematical analysis, Probability (math.PR), 16. Peace & justice, configuration model, Exponential function, Contact process, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 60K35, 82C22, Exponent, Statistics, Probability and Uncertainty, Constant (mathematics), Mathematics - Probability
Abstract

We study the contact process on the configuration model with a power law degree distribution, when the exponent is smaller than or equal to two. We prove that the extinction time grows exponentially fast with the size of the graph and prove two metastability results. First the extinction time divided by its mean converges in distribution toward an exponential random variable with mean one, when the size of the graph tends to infinity. Moreover, the density of infected sites taken at exponential times converges in probability to a constant. This extends previous results in the case of an exponent larger than $2$ obtained in \cite{CD,MMVY,MVY}., Proposition 6.2 replaced by a weaker version (after a gap in its proof was mentioned to us by Daniel Valesin). Does not affect the two main theorems of the paper

Published
2014

15. Localization on 4 sites for vertex-reinforced random walks on $\mathbb{Z}$

Author

Anne-Laure Basdevant, Arvind Singh, Bruno Schapira, Modélisation aléatoire de Paris X (MODAL'X), Université Paris Nanterre (UPN), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)

Subjects
Statistics and Probability, Vertex (graph theory), Probability (math.PR), Order (ring theory), 16. Peace & justice, Random walk, Reinforced random walk, urn model, localization, Combinatorics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 60K35, 60J17, 60J20, FOS: Mathematics, Almost surely, Statistics, Probability and Uncertainty, Mathematics - Probability, Positive probability, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

We characterize non-decreasing weight functions for which the associated one-dimensional vertex reinforced random walk (VRRW) localizes on 4 sites. A phase transition appears for weights of order $n\log \log n$: for weights growing faster than this rate, the VRRW localizes almost surely on at most 4 sites whereas for weights growing slower, the VRRW cannot localize on less than 5 sites. When $w$ is of order $n\log \log n$, the VRRW localizes almost surely on either 4 or 5 sites, both events happening with positive probability., Minor modifications

Published
2014
Full Text
View/download PDF

16. Martingale defocusing and transience of a self-interacting random walk

Author

Perla Sousi, Yuval Peres, Bruno Schapira, Apollo - University of Cambridge Repository, Theory Group - Microsoft Research, Microsoft Research, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Department of Pure Mathematics and Mathematical Statistics (DPMMS), Faculty of mathematics Centre for Mathematical Sciences [Cambridge] (CMS), and University of Cambridge [UK] (CAM)-University of Cambridge [UK] (CAM)

Subjects
Statistics and Probability, Heterogeneous random walk in one dimension, Self-interacting random walk, 010102 general mathematics, Loop-erased random walk, 4901 Applied Mathematics, Probability (math.PR), Random walk, 01 natural sciences, Excited random walk, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Combinatorics, 010104 statistics & probability, 4905 Statistics, Martingale, Mathematics::Probability, 60K35, 49 Mathematical Sciences, Transience, FOS: Mathematics, 0101 mathematics, Statistics, Probability and Uncertainty, Martingale (probability theory), Mathematics - Probability, Mathematics
Abstract

Suppose that $(X,Y,Z)$ is a random walk in $\Z^3$ that moves in the following way: on the first visit to a vertex only $Z$ changes by $\pm 1$ equally likely, while on later visits to the same vertex $(X,Y)$ performs a two-dimensional random walk step. We show that this walk is transient thus answering a question of Benjamini, Kozma and Schapira. One important ingredient of the proof is a dispersion result for martingales.; Supposons que (X, Y, Z) soit une marche aléatoire dans Z3 qui se déplace de la façon suivante : à la première visite en un site, seule la coordonnée Z saute de ±1 avec probabilité uniforme, et aux visites suivantes en ce site (X, Y ) effectue un saut dans l’ensemble {(±1, 0), (0, ±1)} avec probabilité uniforme. Nous montrons que cette marche est transiente, répondant ainsi à une question de Benjamini, Kozma et Schapira. Un ingrédient important de la preuve est un résultat de dispersion pour les martingales.

Published
2014
Full Text
View/download PDF

17. The quenched limiting distributions of a one-dimensional random walk in random scenery

Author

Bruno Schapira, Yueyun Hu, Nadine Guillotin-Plantard, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Laboratoire d'Analyse, Topologie, Probabilités (LATP), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), and Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Random walk in random scenery, 01 natural sciences, 010104 statistics & probability, Mathematics::Probability, 60F05, Strassen algorithm, FOS: Mathematics, 0101 mathematics, Mathematics, Discrete mathematics, Law of the iterated logarithm, Heterogeneous random walk in one dimension, 010102 general mathematics, Probability (math.PR), Contrast (statistics), Limiting, Random walk, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Brownian motion in Brownian Scenery, Weak limit theorem, Statistics, Probability and Uncertainty, Strong approximation, Mathematics - Probability, 60G52
Abstract

International audience; For a one-dimensional random walk in random scenery (RWRS) on Z, we determine its quenched weak limits by applying Strassen's functional law of the iterated logarithm. As a consequence, conditioned on the random scenery, the one-dimensional RWRS does not converge in law, in contrast with the multi-dimensional case.

Published
2013
Full Text
View/download PDF

18. On the one-sided exit problem for stable processes in random scenery

Author

Bruno Schapira, Françoise Pène, Fabienne Castell, Nadine Guillotin-Plantard, Laboratoire d'Analyse, Topologie, Probabilités (LATP), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de mathématiques de Brest (LM), Université de Brest (UBO)-Institut Brestois du Numérique et des Mathématiques (IBNM), Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, 01 natural sciences, Lévy process, Stable process, Combinatorics, 010104 statistics & probability, Mathematics::Probability, 60F05, Survival exponent, 0101 mathematics, Majumdar, Brownian motion, Mathematics, One-sided exit problem, 010102 general mathematics, First passage time, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], One-sided barrier problem, Random scenery, 60K37, 60F17, One sided, 60G18, 60G15, Statistics, Probability and Uncertainty, First-hitting-time model
Abstract

International audience; We consider the one-sided exit problem for stable LÈvy process in random scenery, that is the asymptotic behaviour for $T$ large of the probability $$\mathbb{P}\Big[ \sup_{t\in[0,T]} \Delta_t \leq 1\Big] $$ where $$\Delta_t = \int_{\mathbb{R}} L_t(x) \, dW(x).$$ Here $W=(W(x))_{x\in\mathbb{R}}$ is a two-sided standard real Brownian motion and $(L_t(x))_{x\in\mathbb{R},t\geq 0}$ the local time of a stable Lévy process with index $\alpha\in (1,2]$, independent from the process $W$. Our result confirms some physicists prediction by Redner and Majumdar.

Published
2013
Full Text
View/download PDF

19. On the local time of random processes in random scenery

Author

Françoise Pène, Fabienne Castell, Bruno Schapira, Nadine Guillotin-Plantard, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Probabilités, statistique, physique mathématique (PSPM), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Laboratoire de mathématiques de Brest (LM), Université de Brest (UBO)-Institut Brestois du Numérique et des Mathématiques (IBNM), Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Random walk in random scenery, 60F05, 60F17, 60G15, 60G18, 60K37, Hölder condition, 01 natural sciences, Combinatorics, 010104 statistics & probability, local time, Mathematics::Probability, FOS: Mathematics, 0101 mathematics, Mathematics, Discrete mathematics, local limit theorem, Stochastic process, 010102 general mathematics, Probability (math.PR), Order (ring theory), Random walk, level sets, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Distribution (mathematics), Square-integrable function, Hausdorff dimension, Statistics, Probability and Uncertainty, Random variable, Mathematics - Probability
Abstract

Random walks in random scenery are processes defined by $Z_{n}:=\sum_{k=1}^{n}\xi_{X_{1}+\cdots+X_{k}}$, where basically $(X_{k},k\ge1)$ and $(\xi_{y},y\in\mathbb{Z})$ are two independent sequences of i.i.d. random variables. We assume here that $X_{1}$ is $\mathbb{Z}$-valued, centered and with finite moments of all orders. We also assume that $\xi_{0}$ is $\mathbb{Z}$-valued, centered and square integrable. In this case H. Kesten and F. Spitzer proved that $(n^{-3/4}Z_{[nt]},t\ge0)$ converges in distribution as $n\to\infty$ toward some self-similar process $(\Delta_{t},t\ge0)$ called Brownian motion in random scenery. In a previous paper, we established that $\mathbb{P}(Z_{n}=0)$ behaves asymptotically like a constant times $n^{-3/4}$, as $n\to\infty$. We extend here this local limit theorem: we give a precise asymptotic result for the probability for $Z$ to return to zero simultaneously at several times. As a byproduct of our computations, we show that $\Delta$ admits a bi-continuous version of its local time process which is locally Hölder continuous of order $1/4-\delta$ and $1/6-\delta$, respectively, in the time and space variables, for any $\delta>0$. In particular, this gives a new proof of the fact, previously obtained by Khoshnevisan, that the level sets of $\Delta$ have Hausdorff dimension a.s. equal to $1/4$. We also get the convergence of every moment of the normalized local time of $Z$ toward its continuous counterpart.

Published
2012

20. A $0$-$1$ law for vertex-reinforced random walks on $\mathbb{Z}$ with weight of order $k^\alpha$, $\alpha\in[0,1/2)$

Author

Bruno Schapira

Subjects
Statistics and Probability, Discrete mathematics, Vertex (graph theory), Self-interacting random walk, Order (ring theory), Reinforced random walk, Random walk, Combinatorics, Alpha (programming language), 60K35, K-alpha, $0$-$1$ law, Almost surely, Statistics, Probability and Uncertainty, 60F20, Mathematics
Abstract

We prove that Vertex Reinforced Random Walk on $\mathbb{Z}$ with weight of order $k^\alpha$, with $\alpha\in [0,1/2)$, is either almost surely recurrent or almost surely transient. This improves a previous result of Volkov who showed that the set of sites which are visited infinitely often was a.s. either empty or infinite.

Published
2012
Full Text
View/download PDF

21. A local limit theorem for random walks in random scenery and on randomly oriented lattices

Author

Bruno Schapira, Nadine Guillotin-Plantard, Françoise Pène, Fabienne Castell, Laboratoire d'Analyse, Topologie, Probabilités (LATP), Université Paul Cézanne - Aix-Marseille 3-Université de Provence - Aix-Marseille 1-Centre National de la Recherche Scientifique (CNRS), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de mathématiques de Brest (LM), Université de Brest (UBO)-Institut Brestois du Numérique et des Mathématiques (IBNM), Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), ANR-07-BLAN-0231,MEMEMO,Marches aléatoires, milieux aléatoires, renforcement(2007), Laboratoire d'Analyse, Topologie, Probabilités ( LATP ), Université Paul Cézanne - Aix-Marseille 3-Université de Provence - Aix-Marseille 1-Centre National de la Recherche Scientifique ( CNRS ), Université Claude Bernard Lyon 1 ( UCBL ), Institut Camille Jordan [Villeurbanne] ( ICJ ), École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de mathématiques de Brest ( LM ), Université de Brest ( UBO ) -Institut Brestois du Numérique et des Mathématiques ( IBNM ), Université de Brest ( UBO ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques d'Orsay ( LM-Orsay ), Université Paris-Sud - Paris 11 ( UP11 ) -Centre National de la Recherche Scientifique ( CNRS ), ANR MEMEMO and RANDYMECA,ANR MEMEMO and RANDYMECA, Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), and Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Random walk in random scenery, random walk on randomly oriented lattices, 01 natural sciences, stable process, Stable process, Combinatorics, 010104 statistics & probability, Mathematics::Probability, 60F05, FOS: Mathematics, Limit (mathematics), 0101 mathematics, Mathematics, local limit theorem, Probability (math.PR), 010102 general mathematics, Random walk, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 60G52, Convergence of random variables, Domain (ring theory), Statistics, Probability and Uncertainty, Random variable, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Mathematics - Probability
Abstract

International audience; Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(\xi_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index $\alpha\in (0,2]$ and $\beta\in (0,2]$ respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when $\alpha\neq 1$ and as $n\to \infty$, of $n^{-\delta}Z_n$, for some suitable $\delta>0$ depending on $\alpha$ and $\beta$. Here we are interested in the convergence, as $n\to \infty$, of $n^\delta{\mathbb P}(Z_n=\lfloor n^{\delta} x\rfloor)$, when $x\in \RR$ is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.

Published
2011
Full Text
View/download PDF

22. Internal DLA generated by cookie random walks on $\mathbb{Z}$

Author

Olivier Raimond, Bruno Schapira, Modélisation aléatoire de Paris X (MODAL'X), Université Paris Nanterre (UPN), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), and Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Boundary (topology), 01 natural sciences, cookie random walks, 010104 statistics & probability, 60F15, 60K35, Dimension (vector space), Mathematics::Probability, Law of large numbers, FOS: Mathematics, 0101 mathematics, Mathematics, Discrete mathematics, Heterogeneous random walk in one dimension, law of large numbers, 010102 general mathematics, Probability (math.PR), 16. Peace & justice, Random walk, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], excited random walk, Statistics, Probability and Uncertainty, Mathematics - Probability, Internal DLA
Abstract

We prove a law of large numbers for the right boundary in the model of internal DLA generated by cookie random walks in dimension one. The proof is based on stochastic algorithms techniques., Comment: 8pp

Published
2011
Full Text
View/download PDF

23. Random walk on a building of type Ãr and Brownian motion of the Weyl chamber

Author

Bruno Schapira

Subjects
Statistics and Probability, Pure mathematics, 60B10, Rank (linear algebra), Discretization, Geometry, Random walk, Type (model theory), Affine building, 05C25, Probability theory, 60J25, Symmetric space, GUE process, 60J35, 60C05, 60J10, Affine transformation, Root systems, Statistics, Probability and Uncertainty, 60B15, Brownian motion, 60J60, Mathematics
Abstract

In this paper we study a random walk on an affine building of type Ãr, whose radial part, when suitably normalized, converges toward the Brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane (Probab. Theory Related Fields 89 (1991) 117–129). This extends also the link at the probabilistic level between Riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol and Jeulin in rank one (C. R. Acad. Sci. Paris Sér. I Math. 333 (2001) 785–790). The main ingredients of the proof are a combinatorial formula on the building and the estimate of the transition density proved in Anker et al. (2006).

Published
2009
Full Text
View/download PDF

24. On some generalized reinforced random walk on integers

Author

Olivier Raimond and Bruno Schapira

Subjects
Statistics and Probability, Discrete mathematics, Phase transition, Heterogeneous random walk in one dimension, Function (mathematics), urn processes, Edge (geometry), Random walk, Combinatorics, Mathematics::Probability, Reinforced random walks, Statistics, Probability and Uncertainty, 60F20, Mathematics
Abstract

We consider Reinforced Random Walks where transitions probabilities are a function of the proportions of times the walk has traversed an edge. We give conditions for recurrence or transience. A phase transition is observed, similar to Pemantle [7] on trees

Published
2009
Full Text
View/download PDF

25. Windings of planar random walks and averaged Dehn function

Author

Bruno Schapira, Robert Young, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Institut des Hautes Etudes Scientifiques (IHES), and IHES

Subjects
Statistics and Probability, Group Theory (math.GR), Expected value, winding number, 01 natural sciences, Square (algebra), [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR], Dehn function, averaged Dehn function, Mathematics - Geometric Topology, 010104 statistics & probability, Probability theory, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], FOS: Mathematics, Hexagonal lattice, 60D05, 0101 mathematics, Mathematics, 010102 general mathematics, Mathematical analysis, Winding number, Probability (math.PR), Random function, Geometric Topology (math.GT), Random walk, 52C45, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], simple random walk, Statistics, Probability and Uncertainty, Mathematics - Group Theory, Mathematics - Probability
Abstract

We prove a sharp estimate on the expected value of the integral of the index of a simple random walk on the square or triangular lattice. This gives new lower bounds on the averaged Dehn function, which measures the expected area needed to fill a random curve with a disc.

Published
2008
Full Text
View/download PDF

Catalog

Books, media, physical & digital resources
See catalog results