In this paper, we prove absence of temperature chaos for the two-dimensional discrete Gaussian free field using the convergence of the full extremal process, which has been obtained recently by Biskup and Louidor. This means that the overlap of two points chosen under Gibbs measures at different temperatures has a nontrivial distribution. Whereas this distribution is the same as for the random energy model when the two points are sampled at the same temperature, we point out here that they are different when temperatures are distinct: more precisely, we prove that the mean overlap of two points chosen under Gibbs measures at different temperatures for the DGFF is strictly smaller than the REM's one. Therefore, although neither of these models exhibits temperature chaos, one could say that the DGFF is more chaotic in temperature than the REM.
{"title":"Two-temperatures overlap distribution for the 2D discrete Gaussian free field","authors":"Michel Pain, Olivier Zindy","doi":"10.1214/20-AIHP1091","DOIUrl":"https://doi.org/10.1214/20-AIHP1091","url":null,"abstract":"In this paper, we prove absence of temperature chaos for the two-dimensional discrete Gaussian free field using the convergence of the full extremal process, which has been obtained recently by Biskup and Louidor. This means that the overlap of two points chosen under Gibbs measures at different temperatures has a nontrivial distribution. Whereas this distribution is the same as for the random energy model when the two points are sampled at the same temperature, we point out here that they are different when temperatures are distinct: more precisely, we prove that the mean overlap of two points chosen under Gibbs measures at different temperatures for the DGFF is strictly smaller than the REM's one. Therefore, although neither of these models exhibits temperature chaos, one could say that the DGFF is more chaotic in temperature than the REM.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"68 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72514685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The transition from a weak-disorder (diffusive) to a strong-disorder (localized) phase for directed polymers in a random environment is a well studied phenomenon. In the most common setup, it is established that the phase transition is trivial when the transversal dimension d equals 1 or 2 (the diffusive phase is reduced to β=0) while when d≥3, there is a critical temperature βc∈(0,∞) which delimits the two phases. The proof of the existence of a diffusive regime for d≥3 is based on a second moment method (Comm. Math. Phys. 123 (1989) 529–534, Ann. Probab. 34 (2006) 1746–1770, J. Stat. Phys. 52 (1988) 609–626), and thus relies heavily on the assumption that the variable which encodes the disorder intensity (which in most of the mathematics literature assumes the form eβηx), has finite second moment. The aim of this work is to investigate how the presence/absence of phase transition may depend on the dimension d in the case when the disorder variable displays a heavier tail. To this end we replace eβηx by (1+βωx) where ωx is in the domain of attraction of a stable law with parameter γ∈(1,2). In this setup we show that a non-trivial phase transition occurs if and only if γ>1+2/d. More precisely, when γ≤1+2/d, the free energy of the system is smaller than its annealed counterpart at every temperature whereas when γ>1+2/d the martingale sequence of renormalized partition functions converges to an almost surely positive random variable for all β sufficiently small.
{"title":"Directed polymer in γ-stable random environments","authors":"Robert D. Viveros","doi":"10.1214/20-AIHP1108","DOIUrl":"https://doi.org/10.1214/20-AIHP1108","url":null,"abstract":"The transition from a weak-disorder (diffusive) to a strong-disorder (localized) phase for directed polymers in a random environment is a well studied phenomenon. In the most common setup, it is established that the phase transition is trivial when the transversal dimension d equals 1 or 2 (the diffusive phase is reduced to β=0) while when d≥3, there is a critical temperature βc∈(0,∞) which delimits the two phases. The proof of the existence of a diffusive regime for d≥3 is based on a second moment method (Comm. Math. Phys. 123 (1989) 529–534, Ann. Probab. 34 (2006) 1746–1770, J. Stat. Phys. 52 (1988) 609–626), and thus relies heavily on the assumption that the variable which encodes the disorder intensity (which in most of the mathematics literature assumes the form eβηx), has finite second moment. The aim of this work is to investigate how the presence/absence of phase transition may depend on the dimension d in the case when the disorder variable displays a heavier tail. To this end we replace eβηx by (1+βωx) where ωx is in the domain of attraction of a stable law with parameter γ∈(1,2). In this setup we show that a non-trivial phase transition occurs if and only if γ>1+2/d. More precisely, when γ≤1+2/d, the free energy of the system is smaller than its annealed counterpart at every temperature whereas when γ>1+2/d the martingale sequence of renormalized partition functions converges to an almost surely positive random variable for all β sufficiently small.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"5 1","pages":"1081-1102"},"PeriodicalIF":1.5,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74959065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the discrete directed polymer model with i.i.d. environment and we study the fluctuations 8 of the tail n(W∞ − Wn) of the normalized partition function. It was proven by Comets and Liu [18], that 9 for sufficiently high temperature, the fluctuations converge in distribution towards the product of the limiting par10 tition function and an independent Gaussian random variable. We extend the result to the whole L-region, which 11 is predicted to be the maximal high-temperature region where the Gaussian fluctuations should occur under the 12 considered scaling. To do so, we manage to avoid the heavy 4th-moment computation and instead rely on the local 13 limit theorem for polymers [47, 49] and homogenization. 14 MSC 2010 subject classifications: Primary 60K37, 60K37; secondary 60F05. 15
{"title":"Gaussian fluctuations for the directed polymer partition function in dimension d≥3 and in the whole L2-region","authors":"Clément Cosco, S. Nakajima","doi":"10.1214/20-AIHP1100","DOIUrl":"https://doi.org/10.1214/20-AIHP1100","url":null,"abstract":"We consider the discrete directed polymer model with i.i.d. environment and we study the fluctuations 8 of the tail n(W∞ − Wn) of the normalized partition function. It was proven by Comets and Liu [18], that 9 for sufficiently high temperature, the fluctuations converge in distribution towards the product of the limiting par10 tition function and an independent Gaussian random variable. We extend the result to the whole L-region, which 11 is predicted to be the maximal high-temperature region where the Gaussian fluctuations should occur under the 12 considered scaling. To do so, we manage to avoid the heavy 4th-moment computation and instead rely on the local 13 limit theorem for polymers [47, 49] and homogenization. 14 MSC 2010 subject classifications: Primary 60K37, 60K37; secondary 60F05. 15","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"7 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88563488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A version of the Schelling model on Z is defined, where two types of agents are allocated on the sites. An agent prefers to be surrounded by other agents of its own type, and may choose to move if this is not the case. It then sends a request to an agent of opposite type chosen according to some given moving distribution and, if the move is beneficial for both agents, they swap location. We show that certain choices in the dynamics are crucial for the properties of the model. In particular, the model exhibits different asymptotic behavior depending on whether the moving distribution has bounded or unbounded support. Furthermore, the behavior changes if the agents are lazy in the sense that they only swap location if this strictly improves their situation. Generalizations to a version that includes multiple types are discussed. The work provides a rigorous analysis of so called Kawasaki dynamics on an infinite structure with local interactions.
{"title":"The Schelling model on Z","authors":"Maria Deijfen, Timo Vilkas","doi":"10.1214/20-AIHP1096","DOIUrl":"https://doi.org/10.1214/20-AIHP1096","url":null,"abstract":"A version of the Schelling model on Z is defined, where two types of agents are allocated on the sites. An agent prefers to be surrounded by other agents of its own type, and may choose to move if this is not the case. It then sends a request to an agent of opposite type chosen according to some given moving distribution and, if the move is beneficial for both agents, they swap location. We show that certain choices in the dynamics are crucial for the properties of the model. In particular, the model exhibits different asymptotic behavior depending on whether the moving distribution has bounded or unbounded support. Furthermore, the behavior changes if the agents are lazy in the sense that they only swap location if this strictly improves their situation. Generalizations to a version that includes multiple types are discussed. The work provides a rigorous analysis of so called Kawasaki dynamics on an infinite structure with local interactions.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"160 1","pages":"800-814"},"PeriodicalIF":1.5,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74908313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a unified study of three statistical settings by widening the ρ-estimation method developed in [BBS17]. More specifically, we aim at estimating a density, a hazard rate (from censored data), and a transition intensity of a time inhomogeneous Markov process. We relate the performance of ρ-estimators to deviations of a (possibly unbounded) empirical process. We deduce non-asymptotic risk bounds for an Hellinger-type loss on possibly random models. When the models are convex, maximum likelihood estimators coincide with ρ-estimators, and satisfy therefore our risk bounds. However, our results also apply to some models where the maximum likelihood method does not work. Besides, the robustness properties of ρ-estimators are not, in general, shared by maximum likelihood estimators. Subsequently, we present an alternative procedure to ρ-estimation, more numerically friendly, that yields a piecewise polynomial estimator. We prove theoretical results and carry out some numerical simulations that show the benefits of our approach compared with a more classical one based on maximum likelihood.
{"title":"Estimating a density, a hazard rate, and a transition intensity via the ρ-estimation method","authors":"M. Sart","doi":"10.1214/20-AIHP1076","DOIUrl":"https://doi.org/10.1214/20-AIHP1076","url":null,"abstract":"We propose a unified study of three statistical settings by widening the ρ-estimation method developed in [BBS17]. More specifically, we aim at estimating a density, a hazard rate (from censored data), and a transition intensity of a time inhomogeneous Markov process. We relate the performance of ρ-estimators to deviations of a (possibly unbounded) empirical process. We deduce non-asymptotic risk bounds for an Hellinger-type loss on possibly random models. When the models are convex, maximum likelihood estimators coincide with ρ-estimators, and satisfy therefore our risk bounds. However, our results also apply to some models where the maximum likelihood method does not work. Besides, the robustness properties of ρ-estimators are not, in general, shared by maximum likelihood estimators. Subsequently, we present an alternative procedure to ρ-estimation, more numerically friendly, that yields a piecewise polynomial estimator. We prove theoretical results and carry out some numerical simulations that show the benefits of our approach compared with a more classical one based on maximum likelihood.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"121 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81786249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Global observables for RW: Law of large numbers","authors":"D. Dolgopyat, M. Lenci, Péter Nándori","doi":"10.1214/20-AIHP1072","DOIUrl":"https://doi.org/10.1214/20-AIHP1072","url":null,"abstract":"","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"25 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82535496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a Galton–Watson tree with offspring distribution ν of finite mean. The uniform measure on the boundary of the tree is obtained by putting mass 1 on each vertex of the n-th generation and taking the limit n → ∞. In the case E[ν log(ν)] < ∞, this measure has been well studied, and it is known that the Hausdorff dimension of the measure is equal to log(m) ([3], [14]). When E[ν log(ν)] = ∞, we show that the dimension drops to 0. This answers a question of Lyons, Pemantle and Peres [15] .
{"title":"Hausdorff dimension of the uniform measure of Galton–Watson trees without the XlogX condition","authors":"EF Elie Aidékon","doi":"10.1214/19-aihp1031","DOIUrl":"https://doi.org/10.1214/19-aihp1031","url":null,"abstract":"We consider a Galton–Watson tree with offspring distribution ν of finite mean. The uniform measure on the boundary of the tree is obtained by putting mass 1 on each vertex of the n-th generation and taking the limit n → ∞. In the case E[ν log(ν)] < ∞, this measure has been well studied, and it is known that the Hausdorff dimension of the measure is equal to log(m) ([3], [14]). When E[ν log(ν)] = ∞, we show that the dimension drops to 0. This answers a question of Lyons, Pemantle and Peres [15] .","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"10 1","pages":"2301-2306"},"PeriodicalIF":1.5,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79670945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study existence and uniqueness to multidimensional Reflected Backward Stochastic Differential Equations in an open convex domain, allowing for oblique directions of reflection. In a Markovian framework, combining a priori estimates for penalised equations and compactness arguments, we obtain existence results under quite weak assumptions on the driver of the BSDEs and the direction of reflection, which is allowed to depend on both Y and Z. In a non Markovian framework, we obtain existence and uniqueness result for direction of reflection depending on time and Y. We make use in this case of stability estimates that require some smoothness conditions on the domain and the direction of reflection.
{"title":"Obliquely reflected backward stochastic differential equations","authors":"J. Chassagneux, A. Richou","doi":"10.1214/20-aihp1061","DOIUrl":"https://doi.org/10.1214/20-aihp1061","url":null,"abstract":"In this paper, we study existence and uniqueness to multidimensional Reflected Backward Stochastic Differential Equations in an open convex domain, allowing for oblique directions of reflection. In a Markovian framework, combining a priori estimates for penalised equations and compactness arguments, we obtain existence results under quite weak assumptions on the driver of the BSDEs and the direction of reflection, which is allowed to depend on both Y and Z. In a non Markovian framework, we obtain existence and uniqueness result for direction of reflection depending on time and Y. We make use in this case of stability estimates that require some smoothness conditions on the domain and the direction of reflection.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"35 1","pages":"2868-2896"},"PeriodicalIF":1.5,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89883481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Superdi usions corresponding to di erential operators of the form Lu+βu−αu with mass creation (potential) terms β(·) that are `large functions' are studied. Our construction for superdi usions with large mass creations works for the branching mechanism βu−αu , 0 < γ < 1, as well. Let D ⊆ R be a domain in R. When β is large, the generalized principal eigenvalue λc of L+β in D is typically in nite. Let {Tt, t ≥ 0} denote the Schrödinger semigroup of L + β in D with zero Dirichlet boundary condition. Under the mild assumption that there exists an 0 < h ∈ C(D) so that Tth is nite-valued for all t ≥ 0, we show that there is a uniqueMloc(D)-valued Markov process that satis es a log-Laplace equation in terms of the minimal nonnegative solution to a semilinear initial value problem. Although for super-Brownian motion (SBM) this assumption requires β to be less than quadratic, the quadratic case will be treated as well. When λc = ∞, the usual machinery, including martingale methods and PDE as well as other similar techniques cease to work e ectively, both for the construction and for the investigation of the large time behavior of superdi usions. In this paper, we develop the following two new techniques for the study of the local/global growth of mass and for the spread of superdi usions: • a generalization of the Fleischmann-Swart `Poisson-coupling,' linking superprocesses with branching di usions; • the introduction of a new concept: the `p-generalized principal eigenvalue.' The precise growth rate for the total population of SBM with α(x) = β(x) = 1 + |x| for p ∈ [0, 2] is given in this paper.
{"title":"Superdiffusions with super-exponential growth: Construction, mass and spread","authors":"Zhen-Qing Chen, J. Engländer","doi":"10.1214/19-aihp1018","DOIUrl":"https://doi.org/10.1214/19-aihp1018","url":null,"abstract":"Superdi usions corresponding to di erential operators of the form Lu+βu−αu with mass creation (potential) terms β(·) that are `large functions' are studied. Our construction for superdi usions with large mass creations works for the branching mechanism βu−αu , 0 < γ < 1, as well. Let D ⊆ R be a domain in R. When β is large, the generalized principal eigenvalue λc of L+β in D is typically in nite. Let {Tt, t ≥ 0} denote the Schrödinger semigroup of L + β in D with zero Dirichlet boundary condition. Under the mild assumption that there exists an 0 < h ∈ C(D) so that Tth is nite-valued for all t ≥ 0, we show that there is a uniqueMloc(D)-valued Markov process that satis es a log-Laplace equation in terms of the minimal nonnegative solution to a semilinear initial value problem. Although for super-Brownian motion (SBM) this assumption requires β to be less than quadratic, the quadratic case will be treated as well. When λc = ∞, the usual machinery, including martingale methods and PDE as well as other similar techniques cease to work e ectively, both for the construction and for the investigation of the large time behavior of superdi usions. In this paper, we develop the following two new techniques for the study of the local/global growth of mass and for the spread of superdi usions: • a generalization of the Fleischmann-Swart `Poisson-coupling,' linking superprocesses with branching di usions; • the introduction of a new concept: the `p-generalized principal eigenvalue.' The precise growth rate for the total population of SBM with α(x) = β(x) = 1 + |x| for p ∈ [0, 2] is given in this paper.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"1 1","pages":"1809-1840"},"PeriodicalIF":1.5,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87634081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain Hölder-type classes in which a random field is treated as a space-time function taking values in L-space of random variables. A modified stochastic parabolicity condition involving p is proposed to ensure the finiteness of the associated norm of the solution, which is showed to be sharp by examples. The Schauder-type estimates and the solvability theorem are proved.
{"title":"Stochastic Hölder continuity of random fields governed by a system of stochastic PDEs","authors":"Kai Du, Jiakun Liu, Fu Zhang","doi":"10.1214/19-aihp1000","DOIUrl":"https://doi.org/10.1214/19-aihp1000","url":null,"abstract":"This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain Hölder-type classes in which a random field is treated as a space-time function taking values in L-space of random variables. A modified stochastic parabolicity condition involving p is proposed to ensure the finiteness of the associated norm of the solution, which is showed to be sharp by examples. The Schauder-type estimates and the solvability theorem are proved.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"2 1","pages":"1230-1250"},"PeriodicalIF":1.5,"publicationDate":"2020-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74832940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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