{"title":"Joint localization of directed polymers","authors":"Yurii Bakhtin, Douglas Dow","doi":"10.1214/23-ejp1000","DOIUrl":"https://doi.org/10.1214/23-ejp1000","url":null,"abstract":"","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43196308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce and study a notion of duality for two classes of optimization problems commonly occurring in probability theory. That is, on an abstract measurable space (Ω,F), we consider pairs (E,G) where E is an equivalence relation on Ω and G is a sub-σ-algebra of F; we say that (E,G) satisfies “strong duality” if E is (F⊗F)-measurable and if for all probability measures P,P′ on (Ω,F) we have maxA∈G|P(A)−P′(A)|=minP˜∈Π(P,P′)(1−P˜(E)), where Π(P,P′) denotes the space of couplings of P and P′, and where “max” and “min” assert that the supremum and infimum are in fact achieved. The results herein give wide sufficient conditions for strong duality to hold, thereby extending a form of Kantorovich duality to a class of cost functions which are irregular from the point of view of topology but regular from the point of view of descriptive set theory. The given conditions recover or strengthen classical results, and they have novel consequences in stochastic calculus, point process theory, and random sequence simulation.
{"title":"A strong duality principle for equivalence couplings and total variation","authors":"Adam Quinn Jaffe","doi":"10.1214/23-ejp1016","DOIUrl":"https://doi.org/10.1214/23-ejp1016","url":null,"abstract":"We introduce and study a notion of duality for two classes of optimization problems commonly occurring in probability theory. That is, on an abstract measurable space (Ω,F), we consider pairs (E,G) where E is an equivalence relation on Ω and G is a sub-σ-algebra of F; we say that (E,G) satisfies “strong duality” if E is (F⊗F)-measurable and if for all probability measures P,P′ on (Ω,F) we have maxA∈G|P(A)−P′(A)|=minP˜∈Π(P,P′)(1−P˜(E)), where Π(P,P′) denotes the space of couplings of P and P′, and where “max” and “min” assert that the supremum and infimum are in fact achieved. The results herein give wide sufficient conditions for strong duality to hold, thereby extending a form of Kantorovich duality to a class of cost functions which are irregular from the point of view of topology but regular from the point of view of descriptive set theory. The given conditions recover or strengthen classical results, and they have novel consequences in stochastic calculus, point process theory, and random sequence simulation.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135448136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive quantitative criteria for the existence of density for stochastic line integrals and iterated line integrals along solutions of hypoelliptic differential equations driven by fractional Brownian motion. As an application, we also study the signature uniqueness problem for these rough differential equations.
{"title":"Non-degeneracy of stochastic line integrals","authors":"Xi Geng, Sheng Wang","doi":"10.1214/23-ejp1017","DOIUrl":"https://doi.org/10.1214/23-ejp1017","url":null,"abstract":"We derive quantitative criteria for the existence of density for stochastic line integrals and iterated line integrals along solutions of hypoelliptic differential equations driven by fractional Brownian motion. As an application, we also study the signature uniqueness problem for these rough differential equations.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135211237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Several terms in an asymptotic estimate for the renewal mass function in a discrete random walk which has positive mean and regularly varying right-hand tail are given. Similar results are given for the renewal density in the absolutely continuous case.
{"title":"Local behaviour of the remainder in Renewal theory","authors":"Ron Doney","doi":"10.1214/23-ejp1008","DOIUrl":"https://doi.org/10.1214/23-ejp1008","url":null,"abstract":"Several terms in an asymptotic estimate for the renewal mass function in a discrete random walk which has positive mean and regularly varying right-hand tail are given. Similar results are given for the renewal density in the absolutely continuous case.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135211244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Anatomy of a Gaussian giant: supercritical level-sets of the free field on regular graphs","authors":"Guillaume Conchon--Kerjan","doi":"10.1214/23-ejp920","DOIUrl":"https://doi.org/10.1214/23-ejp920","url":null,"abstract":"","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48572378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by [G. Cannizzaro, M. Hairer, Comm. Pure Applied Math., '22], we provide a construction of the Brownian Web (see [T'oth B., Werner W., Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar, Ann. Probab., '04]), i.e. a family of coalescing Brownian motions starting from every point in $mathbb R^2$, as a random variable taking values in the space of (spatial) $mathbb R$-trees. This gives a stronger topology than the classical one {(i.e. Hausdorff convergence on closed sets of paths)}, thus providing us with more continuous functions of the Brownian Web and ruling out a number of potential pathological behaviours. Along the way, we introduce a modification of the topology of spatial $mathbb R$-trees in [T. Duquesne, J.-F. Le Gall, Probab. Theory Related Fields, '05] and [M. T. Barlow, D. A. Croydon, T. Kumagai, Ann. Probab. '17] which makes it a complete separable metric space and could be of independent interest. We determine some properties of the characterisation of the Brownian Web in this context (e.g. its box-counting dimension) and recover some which were determined in earlier works, such as duality, special points and convergence of the graphical representation of coalescing random walks.
{"title":"The Brownian Web as a random R-tree","authors":"G. Cannizzaro, Martin Hairer","doi":"10.1214/23-ejp984","DOIUrl":"https://doi.org/10.1214/23-ejp984","url":null,"abstract":"Motivated by [G. Cannizzaro, M. Hairer, Comm. Pure Applied Math., '22], we provide a construction of the Brownian Web (see [T'oth B., Werner W., Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar, Ann. Probab., '04]), i.e. a family of coalescing Brownian motions starting from every point in $mathbb R^2$, as a random variable taking values in the space of (spatial) $mathbb R$-trees. This gives a stronger topology than the classical one {(i.e. Hausdorff convergence on closed sets of paths)}, thus providing us with more continuous functions of the Brownian Web and ruling out a number of potential pathological behaviours. Along the way, we introduce a modification of the topology of spatial $mathbb R$-trees in [T. Duquesne, J.-F. Le Gall, Probab. Theory Related Fields, '05] and [M. T. Barlow, D. A. Croydon, T. Kumagai, Ann. Probab. '17] which makes it a complete separable metric space and could be of independent interest. We determine some properties of the characterisation of the Brownian Web in this context (e.g. its box-counting dimension) and recover some which were determined in earlier works, such as duality, special points and convergence of the graphical representation of coalescing random walks.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45629027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jonas Jalowy, Zakhar Kabluchko, Matthias Löwe, Alexander Marynych
We investigate increasing propagation of chaos for the mean-field Ising model of ferromagnetism (also known as the Curie-Weiss model) with N spins at inverse temperature β>0 and subject to an external magnetic field of strength h∈R. Using a different proof technique than in Ben Arous and Zeitouni [Ann. Inst. H. Poincaré: Probab. Statist., 35(1): 85–102, 1999] we confirm the well-known propagation of chaos phenomenon: If k=k(N)=o(N) as N→∞, then the k’th marginal distribution of the Gibbs measure converges to a product measure at β<1 or h≠0 and to a mixture of two product measures, if β>1 and h=0. More importantly, we also show that if k(N)∕N→α∈(0,1], this property is lost and we identify a non-zero limit of the total variation distance between the number of positive spins among any k-tuple and the corresponding binomial distribution.
{"title":"When does the chaos in the Curie-Weiss model stop to propagate?","authors":"Jonas Jalowy, Zakhar Kabluchko, Matthias Löwe, Alexander Marynych","doi":"10.1214/23-ejp1039","DOIUrl":"https://doi.org/10.1214/23-ejp1039","url":null,"abstract":"We investigate increasing propagation of chaos for the mean-field Ising model of ferromagnetism (also known as the Curie-Weiss model) with N spins at inverse temperature β>0 and subject to an external magnetic field of strength h∈R. Using a different proof technique than in Ben Arous and Zeitouni [Ann. Inst. H. Poincaré: Probab. Statist., 35(1): 85–102, 1999] we confirm the well-known propagation of chaos phenomenon: If k=k(N)=o(N) as N→∞, then the k’th marginal distribution of the Gibbs measure converges to a product measure at β<1 or h≠0 and to a mixture of two product measures, if β>1 and h=0. More importantly, we also show that if k(N)∕N→α∈(0,1], this property is lost and we identify a non-zero limit of the total variation distance between the number of positive spins among any k-tuple and the corresponding binomial distribution.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135710346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the Potts model on a two-dimensional periodic rectangular lattice with general coupling constants Jij>0, where i,j∈{1,2,3} are the possible spin values (or colors). The resulting energy landscape is thus significantly more complex than in the original Ising or Potts models. The system evolves according to a Glauber-type spin-flipping dynamics. We focus on a region of the parameter space where there are two symmetric metastable states and a stable state, and the height of a direct path between the metastable states is equal to the height of a direct path between any metastable state and the stable state. We study the metastable transition time in probability and in expectation, the mixing time of the dynamics and the spectral gap of the system when the inverse temperature β tends to infinity. Then, we identify all the critical configurations that are visited with high probability during the metastable transition. Our main tool is the so-called pathwise approach to metastability, which requires a detailed analysis of the energy landscape.
{"title":"Metastability of the three-state Potts model with general interactions","authors":"Gianmarco Bet, Anna Gallo, Seonwoo Kim","doi":"10.1214/23-ejp1003","DOIUrl":"https://doi.org/10.1214/23-ejp1003","url":null,"abstract":"We consider the Potts model on a two-dimensional periodic rectangular lattice with general coupling constants Jij>0, where i,j∈{1,2,3} are the possible spin values (or colors). The resulting energy landscape is thus significantly more complex than in the original Ising or Potts models. The system evolves according to a Glauber-type spin-flipping dynamics. We focus on a region of the parameter space where there are two symmetric metastable states and a stable state, and the height of a direct path between the metastable states is equal to the height of a direct path between any metastable state and the stable state. We study the metastable transition time in probability and in expectation, the mixing time of the dynamics and the spectral gap of the system when the inverse temperature β tends to infinity. Then, we identify all the critical configurations that are visited with high probability during the metastable transition. Our main tool is the so-called pathwise approach to metastability, which requires a detailed analysis of the energy landscape.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135953179","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider generalized inversions and descents in finite Weyl groups. We establish Coxeter-theoretic properties of indicator random variables of positive roots such as the covariance of two such indicator random variables. We then compute the variances of generalized inversions and descents in classical types. We finally use the dependency graph method to prove central limit theorems for general antichains in root posets and in particular for generalized descents, and then for generalized inversions.
{"title":"Central limit theorems for generalized descents and generalized inversions in finite root systems","authors":"Kathrin Meier, Christian Stump","doi":"10.1214/23-ejp1031","DOIUrl":"https://doi.org/10.1214/23-ejp1031","url":null,"abstract":"We consider generalized inversions and descents in finite Weyl groups. We establish Coxeter-theoretic properties of indicator random variables of positive roots such as the covariance of two such indicator random variables. We then compute the variances of generalized inversions and descents in classical types. We finally use the dependency graph method to prove central limit theorems for general antichains in root posets and in particular for generalized descents, and then for generalized inversions.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135212714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson–Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox–Boolean model. The phase transitions are established under individually as well as jointly varying parameters. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in [Hof05].
{"title":"Continuum percolation in a nonstabilizing environment","authors":"Benedikt Jahnel, Sanjoy Kumar Jhawar, Anh Duc Vu","doi":"10.1214/23-ejp1029","DOIUrl":"https://doi.org/10.1214/23-ejp1029","url":null,"abstract":"We prove phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson–Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox–Boolean model. The phase transitions are established under individually as well as jointly varying parameters. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in [Hof05].","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135213402","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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