We provide an existence theory for gradient Gibbs measures for Z-valued spin models on regular trees which are not invariant under translations of the tree, assuming only summability of the transfer operator. The gradient states we obtain are delocalized. The construction we provide for them starts from a two-layer hidden Markov model representation in a setup which is not invariant under tree-automorphisms, involving internal q-spin models. The proofs of existence and lack of translation invariance of infinite-volume gradient states are based on properties of the local pseudo-unstable manifold of the corresponding discrete dynamical systems of these internal models, around the free state, at large q.
{"title":"Existence of gradient Gibbs measures on regular trees which are not translation invariant","authors":"Florian Henning, C. Kuelske","doi":"10.1214/22-aap1883","DOIUrl":"https://doi.org/10.1214/22-aap1883","url":null,"abstract":"We provide an existence theory for gradient Gibbs measures for Z-valued spin models on regular trees which are not invariant under translations of the tree, assuming only summability of the transfer operator. The gradient states we obtain are delocalized. The construction we provide for them starts from a two-layer hidden Markov model representation in a setup which is not invariant under tree-automorphisms, involving internal q-spin models. The proofs of existence and lack of translation invariance of infinite-volume gradient states are based on properties of the local pseudo-unstable manifold of the corresponding discrete dynamical systems of these internal models, around the free state, at large q.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45451503","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 characterize the tail behavior of the distribution of the PageRank of a uniformly chosen vertex in a directed preferential attachment graph and show that it decays as a power law with an explicit exponent that is described in terms of the model parameters. Interestingly, this power law is heavier than the tail of the limiting in-degree distribution, which goes against the commonly accepted power law hypothesis. This deviation from the power law hypothesis points at the structural differences between the inbound neighborhoods of typical vertices in a preferential attachment graph versus those in static random graph models where the power law hypothesis has been proven to hold (e.g., directed configuration models and inhomogeneous random digraphs). In addition to characterizing the PageRank distribution of a typical vertex, we also characterize the explicit growth rate of the PageRank of the oldest vertex as the network size grows.
{"title":"PageRank asymptotics on directed preferential attachment networks","authors":"Sayantan Banerjee, Mariana Olvera-Cravioto","doi":"10.1214/21-aap1757","DOIUrl":"https://doi.org/10.1214/21-aap1757","url":null,"abstract":"We characterize the tail behavior of the distribution of the PageRank of a uniformly chosen vertex in a directed preferential attachment graph and show that it decays as a power law with an explicit exponent that is described in terms of the model parameters. Interestingly, this power law is heavier than the tail of the limiting in-degree distribution, which goes against the commonly accepted power law hypothesis. This deviation from the power law hypothesis points at the structural differences between the inbound neighborhoods of typical vertices in a preferential attachment graph versus those in static random graph models where the power law hypothesis has been proven to hold (e.g., directed configuration models and inhomogeneous random digraphs). In addition to characterizing the PageRank distribution of a typical vertex, we also characterize the explicit growth rate of the PageRank of the oldest vertex as the network size grows.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42209327","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}
Given a sequence of i.i.d. random functions $Psi_{n}:mathbb{R}tomathbb{R}$, $ninmathbb{N}$, we consider the iterated function system and Markov chain which is recursively defined by $X_{0}^{x}:=x$ and $X_{n}^{x}:=Psi_{n-1}(X_{n-1}^{x})$ for $xinmathbb{R}$ and $ninmathbb{N}$. Under the two basic assumptions that the $Psi_{n}$ are a.s. continuous at any point in $mathbb{R}$ and asymptotically linear at the"endpoints"$pminfty$, we study the tail behavior of the stationary laws of such Markov chains by means of Markov renewal theory. Our approach provides an extension of Goldie's implicit renewal theory and can also be viewed as an adaptation of Kesten's work on products of random matrices to one-dimensional function systems as described. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics, e.g. ARCH models and random logistic transforms.
{"title":"Asymptotically linear iterated function systems on the real line","authors":"G. Alsmeyer, S. Brofferio, D. Buraczewski","doi":"10.1214/22-aap1812","DOIUrl":"https://doi.org/10.1214/22-aap1812","url":null,"abstract":"Given a sequence of i.i.d. random functions $Psi_{n}:mathbb{R}tomathbb{R}$, $ninmathbb{N}$, we consider the iterated function system and Markov chain which is recursively defined by $X_{0}^{x}:=x$ and $X_{n}^{x}:=Psi_{n-1}(X_{n-1}^{x})$ for $xinmathbb{R}$ and $ninmathbb{N}$. Under the two basic assumptions that the $Psi_{n}$ are a.s. continuous at any point in $mathbb{R}$ and asymptotically linear at the\"endpoints\"$pminfty$, we study the tail behavior of the stationary laws of such Markov chains by means of Markov renewal theory. Our approach provides an extension of Goldie's implicit renewal theory and can also be viewed as an adaptation of Kesten's work on products of random matrices to one-dimensional function systems as described. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics, e.g. ARCH models and random logistic transforms.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45824368","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 study a class of reinforcement models involving a Poisson process on the vertices of certain infinite graphs G. When a vertex fires, one of the edges incident to that vertex is selected. The edge selection is biased towards edges that have been selected many times previously, and a parameter α governs the strength of this bias. We show that for various graphs (including all graphs of bounded degree), if α 1 (the very strong reinforcement regime) then the random subgraph consisting of edges that are ever selected by this process does not percolate (all connected components are finite). Combined with results appearing in a companion paper, this proves that on these graphs, with α sufficiently large, all connected components are in fact trees. If the Poisson firing rates are constant over the vertices, then these trees are of diameter at most 3. The proof of non-percolation relies on coupling with a percolationtype model that may be of interest in its own right.
{"title":"Absence of WARM percolation in the very strong reinforcement regime","authors":"C. Hirsch, Mark Holmes, V. Kleptsyn","doi":"10.1214/20-AAP1587","DOIUrl":"https://doi.org/10.1214/20-AAP1587","url":null,"abstract":"We study a class of reinforcement models involving a Poisson process on the vertices of certain infinite graphs G. When a vertex fires, one of the edges incident to that vertex is selected. The edge selection is biased towards edges that have been selected many times previously, and a parameter α governs the strength of this bias. We show that for various graphs (including all graphs of bounded degree), if α 1 (the very strong reinforcement regime) then the random subgraph consisting of edges that are ever selected by this process does not percolate (all connected components are finite). Combined with results appearing in a companion paper, this proves that on these graphs, with α sufficiently large, all connected components are in fact trees. If the Poisson firing rates are constant over the vertices, then these trees are of diameter at most 3. The proof of non-percolation relies on coupling with a percolationtype model that may be of interest in its own right.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82176321","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 asymmetric simple exclusion process (ASEP) on Z with initial data such that in the large time particle density ρ(·) a discontinuity (shock) at the origin is created. At the shock, the value of ρ jumps from zero to one, but ρ(−ε), 1−ρ(ε) > 0 for any ε > 0. We are interested in the rescaled position of a tagged particle which enters the shock with positive probability. We show that, inside the shock region, the particle position has the KPZ-typical 1/3 fluctuations, a FGUE×FGUE limit law and a degenerated correlation length. Outside the shock region, the particle fluctuates as if there was no shock. Our arguments are mostly probabilistic, in particular, the mixing times of countable state space ASEPs are instrumental to study the fluctuations at shocks.
{"title":"GUE×GUE limit law at hard shocks in ASEP","authors":"Peter Nejjar","doi":"10.1214/20-AAP1591","DOIUrl":"https://doi.org/10.1214/20-AAP1591","url":null,"abstract":"We consider the asymmetric simple exclusion process (ASEP) on Z with initial data such that in the large time particle density ρ(·) a discontinuity (shock) at the origin is created. At the shock, the value of ρ jumps from zero to one, but ρ(−ε), 1−ρ(ε) > 0 for any ε > 0. We are interested in the rescaled position of a tagged particle which enters the shock with positive probability. We show that, inside the shock region, the particle position has the KPZ-typical 1/3 fluctuations, a FGUE×FGUE limit law and a degenerated correlation length. Outside the shock region, the particle fluctuates as if there was no shock. Our arguments are mostly probabilistic, in particular, the mixing times of countable state space ASEPs are instrumental to study the fluctuations at shocks.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80042115","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 investigate the uniform boundedness of the fronts of the solutions to the randomized Fisher-KPP equation and to its linearization, the parabolic Anderson model. It has been known that for the standard (i.e. deterministic) Fisher-KPP equation, as well as for the special case of a randomized Fisher-KPP equation with so-called ignition type nonlinearity, one has a uniformly bounded (in time) transition front. Here, we show that this property of having a uniformly bounded transition front fails to hold for the general randomized Fisher-KPP equation. Nevertheless, we establish that this property does hold true for the parabolic Anderson model.
{"title":"(Un-)bounded transition fronts for the parabolic Anderson model and the randomized F-KPP equation","authors":"J. vCern'y, Alexander Drewitz, Lars Schmitz","doi":"10.1214/22-aap1869","DOIUrl":"https://doi.org/10.1214/22-aap1869","url":null,"abstract":"We investigate the uniform boundedness of the fronts of the solutions to the randomized Fisher-KPP equation and to its linearization, the parabolic Anderson model. It has been known that for the standard (i.e. deterministic) Fisher-KPP equation, as well as for the special case of a randomized Fisher-KPP equation with so-called ignition type nonlinearity, one has a uniformly bounded (in time) transition front. Here, we show that this property of having a uniformly bounded transition front fails to hold for the general randomized Fisher-KPP equation. Nevertheless, we establish that this property does hold true for the parabolic Anderson model.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42630152","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 majority vote process was one of the first interacting particle systems to be investigated. It can be described briefly as follows. There are two possible opinions at each site of a graph G. At rate 1 − ε, the opinion at a site aligns with the majority opinion at its neighboring sites and, at rate ε, the opinion at a site is randomized due to noise, where ε ∈ [0, 1] is a parameter. Despite the simple dynamics of the majority vote process, its equilibrium behavior is difficult to analyze when the noise rate is small but positive. In particular, when the underlying graph is G = Z with n ≥ 2, it is not known whether the process possesses more than one equilibrium. This is surprising, especially in light of the close analogy between this model and the stochastic Ising model, where much more is known. Here, we study the majority vote process on the infinite tree Td with vertex degree d. For d ≥ 5 and small noise, we show that there are uncountably many mutually singular equilibria, with convergence to such an equilibrium occurring exponentially quickly from nearby initial states. Our methods are quite flexible and extend to a broader class of models, consensus processes. This class includes the stochastic Ising model and other processes in which the dynamics at a site depend on the number of neighbors holding a given opinion. All of our proofs are carried out in this broader context.
{"title":"The majority vote process and other consensus processes on trees","authors":"M. Bramson, L. Gray","doi":"10.1214/20-AAP1586","DOIUrl":"https://doi.org/10.1214/20-AAP1586","url":null,"abstract":"The majority vote process was one of the first interacting particle systems to be investigated. It can be described briefly as follows. There are two possible opinions at each site of a graph G. At rate 1 − ε, the opinion at a site aligns with the majority opinion at its neighboring sites and, at rate ε, the opinion at a site is randomized due to noise, where ε ∈ [0, 1] is a parameter. Despite the simple dynamics of the majority vote process, its equilibrium behavior is difficult to analyze when the noise rate is small but positive. In particular, when the underlying graph is G = Z with n ≥ 2, it is not known whether the process possesses more than one equilibrium. This is surprising, especially in light of the close analogy between this model and the stochastic Ising model, where much more is known. Here, we study the majority vote process on the infinite tree Td with vertex degree d. For d ≥ 5 and small noise, we show that there are uncountably many mutually singular equilibria, with convergence to such an equilibrium occurring exponentially quickly from nearby initial states. Our methods are quite flexible and extend to a broader class of models, consensus processes. This class includes the stochastic Ising model and other processes in which the dynamics at a site depend on the number of neighbors holding a given opinion. All of our proofs are carried out in this broader context.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84302467","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 convergence rate in Wasserstein distance is estimated for the empirical measures of symmetric semilinear SPDEs. Unlike in the finite-dimensional case that the convergence is of algebraic order in time, in the present situation the convergence is of log order with a power given by eigenvalues of the underlying linear operator.
{"title":"Convergence in Wasserstein distance for empirical measures of semilinear SPDEs","authors":"Feng-Yu Wang","doi":"10.1214/22-aap1807","DOIUrl":"https://doi.org/10.1214/22-aap1807","url":null,"abstract":"The convergence rate in Wasserstein distance is estimated for the empirical measures of symmetric semilinear SPDEs. Unlike in the finite-dimensional case that the convergence is of algebraic order in time, in the present situation the convergence is of log order with a power given by eigenvalues of the underlying linear operator.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42965204","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}
Konstantinos Dareiotis, M'at'e Gerencs'er, Khoa Le
We derive sharp strong convergence rates for the Euler-Maruyama scheme approximating multidimensional SDEs with multiplicative noise without imposing any regularity condition on the drift coefficient. In case the noise is additive, we show that Sobolev regularity can be leveraged to obtain improved rate: drifts with regularity of order $alpha in (0,1)$ lead to rate $(1+alpha)/2$.
{"title":"Quantifying a convergence theorem of Gyöngy and Krylov","authors":"Konstantinos Dareiotis, M'at'e Gerencs'er, Khoa Le","doi":"10.1214/22-aap1867","DOIUrl":"https://doi.org/10.1214/22-aap1867","url":null,"abstract":"We derive sharp strong convergence rates for the Euler-Maruyama scheme approximating multidimensional SDEs with multiplicative noise without imposing any regularity condition on the drift coefficient. In case the noise is additive, we show that Sobolev regularity can be leveraged to obtain improved rate: drifts with regularity of order $alpha in (0,1)$ lead to rate $(1+alpha)/2$.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2021-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41824357","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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