We discuss a thinning and an embedding procedure to construct finite Gibbs processes with a given Papangelou intensity. Extending the approach of Hofer-Temmel (Electron. J. Probab. 24 (2019) 1–22) and Hofer-Temmel and Houdebert (Stochastic Process. Appl. 129 (2019) 3922–3940) we will use this to couple two finite Gibbs processes with different boundary conditions. As one application we will establish Poisson approximation of point processes derived from certain infinite volume Gibbs processes via dependent thinning. As another application we shall discuss empty space probabilities of certain Gibbs processes.
{"title":"Disagreement coupling of Gibbs processes with an application to Poisson approximation","authors":"Moritz Otto","doi":"10.1214/22-aap1916","DOIUrl":"https://doi.org/10.1214/22-aap1916","url":null,"abstract":"We discuss a thinning and an embedding procedure to construct finite Gibbs processes with a given Papangelou intensity. Extending the approach of Hofer-Temmel (Electron. J. Probab. 24 (2019) 1–22) and Hofer-Temmel and Houdebert (Stochastic Process. Appl. 129 (2019) 3922–3940) we will use this to couple two finite Gibbs processes with different boundary conditions. As one application we will establish Poisson approximation of point processes derived from certain infinite volume Gibbs processes via dependent thinning. As another application we shall discuss empty space probabilities of certain Gibbs processes.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135948527","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 establish an analytic framework for studying set-valued backward stochastic differential equations (set-valued BSDE), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will make use of the notion of the Hukuhara difference between sets, in order to compensate the lack of “inverse” operation of the traditional Minkowski addition, whence the vector space structure in set-valued analysis. While proving the well-posedness of a class of set-valued BSDEs, we shall also address some fundamental issues regarding generalized Aumann–Itô integrals, especially when it is connected to the martingale representation theorem. In particular, we propose some necessary extensions of the integral that can be used to represent set-valued martingales with nonsingleton initial values. This extension turns out to be essential for the study of set-valued BSDEs.
{"title":"Set-valued backward stochastic differential equations","authors":"Çağın Ararat, Jin Ma, Wenqian Wu","doi":"10.1214/22-aap1896","DOIUrl":"https://doi.org/10.1214/22-aap1896","url":null,"abstract":"In this paper, we establish an analytic framework for studying set-valued backward stochastic differential equations (set-valued BSDE), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will make use of the notion of the Hukuhara difference between sets, in order to compensate the lack of “inverse” operation of the traditional Minkowski addition, whence the vector space structure in set-valued analysis. While proving the well-posedness of a class of set-valued BSDEs, we shall also address some fundamental issues regarding generalized Aumann–Itô integrals, especially when it is connected to the martingale representation theorem. In particular, we propose some necessary extensions of the integral that can be used to represent set-valued martingales with nonsingleton initial values. This extension turns out to be essential for the study of set-valued BSDEs.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136168563","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 are concerned with a mixture of Boltzmann and McKean–Vlasov-type equations, this means (in probabilistic terms) equations with coefficients depending on the law of the solution itself, and driven by a Poisson point measure with the intensity depending also on the law of the solution. Both the analytical Boltzmann equation and the probabilistic interpretation initiated by Tanaka (Z. Wahrsch. Verw. Gebiete 46 (1978/79) 67–105; J. Fac. Sci., Univ. Tokyo, Sect. IA, Math. 34 (1987) 351–369) have intensively been discussed in the literature for specific models related to the behavior of gas molecules. In this paper, we consider general abstract coefficients that may include mean field effects and then we discuss the link with specific models as well. In contrast with the usual approach in which integral equations are used in order to state the problem, we employ here a new formulation of the problem in terms of flows of self-maps on the space of probability measure endowed with the Wasserstein distance. This point of view already appeared in the framework of rough differential equations. Our results concern existence and uniqueness of the solution, in the formulation of flows, but we also prove that the “flow solution” is a solution of the classical integral weak equation and admits a probabilistic interpretation. Moreover, we obtain stability results and regularity with respect to the time for such solutions. Finally we prove the convergence of empirical measures based on particle systems to the solution of our problem, and we obtain the rate of convergence. We discuss as examples the homogeneous and the inhomogeneous Boltzmann (Enskog) equation with hard potentials.
{"title":"Construction of Boltzmann and McKean–Vlasov type flows (the sewing lemma approach)","authors":"Aurélien Alfonsi, Vlad Bally","doi":"10.1214/22-aap1894","DOIUrl":"https://doi.org/10.1214/22-aap1894","url":null,"abstract":"We are concerned with a mixture of Boltzmann and McKean–Vlasov-type equations, this means (in probabilistic terms) equations with coefficients depending on the law of the solution itself, and driven by a Poisson point measure with the intensity depending also on the law of the solution. Both the analytical Boltzmann equation and the probabilistic interpretation initiated by Tanaka (Z. Wahrsch. Verw. Gebiete 46 (1978/79) 67–105; J. Fac. Sci., Univ. Tokyo, Sect. IA, Math. 34 (1987) 351–369) have intensively been discussed in the literature for specific models related to the behavior of gas molecules. In this paper, we consider general abstract coefficients that may include mean field effects and then we discuss the link with specific models as well. In contrast with the usual approach in which integral equations are used in order to state the problem, we employ here a new formulation of the problem in terms of flows of self-maps on the space of probability measure endowed with the Wasserstein distance. This point of view already appeared in the framework of rough differential equations. Our results concern existence and uniqueness of the solution, in the formulation of flows, but we also prove that the “flow solution” is a solution of the classical integral weak equation and admits a probabilistic interpretation. Moreover, we obtain stability results and regularity with respect to the time for such solutions. Finally we prove the convergence of empirical measures based on particle systems to the solution of our problem, and we obtain the rate of convergence. We discuss as examples the homogeneous and the inhomogeneous Boltzmann (Enskog) equation with hard potentials.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119568","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 increasing supermartingale coupling, introduced by Nutz and Stebegg (Ann. Probab. 46 (2018) 3351–3398) is an extreme point of the set of “supermartingale” couplings between two real probability measures in convex-decreasing order. In the present paper we provide an explicit construction of a triple of functions, on the graph of which the increasing supermartingale coupling concentrates. In particular, we show that the increasing supermartingale coupling can be identified with the left-curtain martingale coupling and the antitone coupling to the left and to the right of a uniquely determined regime-switching point, respectively. Our construction is based on the concept of the shadow measure. We show how to determine the potential of the shadow measure associated to a supermartingale, extending the recent results of Beiglböck et al. (Electron. Commun. Probab. 27 (2022) 1–12) obtained in the martingale setting.
{"title":"A potential-based construction of the increasing supermartingale coupling","authors":"Erhan Bayraktar, Shuoqing Deng, Dominykas Norgilas","doi":"10.1214/22-aap1907","DOIUrl":"https://doi.org/10.1214/22-aap1907","url":null,"abstract":"The increasing supermartingale coupling, introduced by Nutz and Stebegg (Ann. Probab. 46 (2018) 3351–3398) is an extreme point of the set of “supermartingale” couplings between two real probability measures in convex-decreasing order. In the present paper we provide an explicit construction of a triple of functions, on the graph of which the increasing supermartingale coupling concentrates. In particular, we show that the increasing supermartingale coupling can be identified with the left-curtain martingale coupling and the antitone coupling to the left and to the right of a uniquely determined regime-switching point, respectively. Our construction is based on the concept of the shadow measure. We show how to determine the potential of the shadow measure associated to a supermartingale, extending the recent results of Beiglböck et al. (Electron. Commun. Probab. 27 (2022) 1–12) obtained in the martingale setting.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119576","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 present normal approximation results at the process level for local functionals defined on dynamic Poisson processes in Rd. The dynamics we study here are those of a Markov birth–death process. We prove functional limit theorems in the so-called thermodynamic regime. Our results are applicable to several functionals of interest in the stochastic geometry literature, including subgraph and component counts in the random geometric graphs.
{"title":"Functional central limit theorems for local statistics of spatial birth–death processes in the thermodynamic regime","authors":"Efe Onaran, Omer Bobrowski, Robert J. Adler","doi":"10.1214/22-aap1912","DOIUrl":"https://doi.org/10.1214/22-aap1912","url":null,"abstract":"We present normal approximation results at the process level for local functionals defined on dynamic Poisson processes in Rd. The dynamics we study here are those of a Markov birth–death process. We prove functional limit theorems in the so-called thermodynamic regime. Our results are applicable to several functionals of interest in the stochastic geometry literature, including subgraph and component counts in the random geometric graphs.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136167558","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}
Jean-Rene Chazottes, Sandro Gallo, Daniel Takahashi
Stochastic chains of unbounded memory (SCUMs) are generalization of Markov chains, also known in the literature as “chains with complete connections” or “g-measures”. We obtain Gaussian concentration bounds (GCB) in this large class of models, for general alphabets, under two different conditions on the kernel: (1) when the sum of its oscillations is less than one, or (2) when the sum of its variations is finite, that is, belongs to ℓ1(N). We also obtain explicit constants as functions of the parameters of the model. Our conditions are sharp in the sense that we exhibit examples of SCUMs that do not have GCB and for which the sum of oscillations is 1+ϵ, or the variation belongs to ℓ1+ϵ(N) for any ϵ>0. These examples are based on the existence of phase transitions. We illustrate our results with four applications. First, we derive a Dvoretzky–Kiefer–Wolfowitz-type inequality which gives a uniform control on the fluctuations of the empirical measure. Second, in the finite-alphabet case, we obtain an upper bound on the d¯-distance between two stationary SCUMs and, as a by-product, we obtain new explicit bounds on the speed of Markovian approximation in d¯. Third, we derive new bounds on the fluctuations of the “plug-in” estimator for entropy. Fourth, we obtain new rate of convergence for the maximum likelihood estimator of conditional probability.
{"title":"Gaussian concentration bounds for stochastic chains of unbounded memory","authors":"Jean-Rene Chazottes, Sandro Gallo, Daniel Takahashi","doi":"10.1214/22-aap1893","DOIUrl":"https://doi.org/10.1214/22-aap1893","url":null,"abstract":"Stochastic chains of unbounded memory (SCUMs) are generalization of Markov chains, also known in the literature as “chains with complete connections” or “g-measures”. We obtain Gaussian concentration bounds (GCB) in this large class of models, for general alphabets, under two different conditions on the kernel: (1) when the sum of its oscillations is less than one, or (2) when the sum of its variations is finite, that is, belongs to ℓ1(N). We also obtain explicit constants as functions of the parameters of the model. Our conditions are sharp in the sense that we exhibit examples of SCUMs that do not have GCB and for which the sum of oscillations is 1+ϵ, or the variation belongs to ℓ1+ϵ(N) for any ϵ>0. These examples are based on the existence of phase transitions. We illustrate our results with four applications. First, we derive a Dvoretzky–Kiefer–Wolfowitz-type inequality which gives a uniform control on the fluctuations of the empirical measure. Second, in the finite-alphabet case, we obtain an upper bound on the d¯-distance between two stationary SCUMs and, as a by-product, we obtain new explicit bounds on the speed of Markovian approximation in d¯. Third, we derive new bounds on the fluctuations of the “plug-in” estimator for entropy. Fourth, we obtain new rate of convergence for the maximum likelihood estimator of conditional probability.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119332","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 derive an anomalous, sub-diffusive scaling limit for a one-dimen-sional version of the Mott random walk. The limiting process can be viewed heuristically as a one-dimensional diffusion with an absolutely continuous speed measure and a discontinuous scale function, as given by a two-sided stable subordinator. Corresponding to intervals of low conductance in the discrete model, the discontinuities in the scale function act as barriers off which the limiting process reflects for some time before crossing. We also discuss how, by incorporating a Bouchaud trap model element into the setting, it is possible to combine this “blocking” mechanism with one of “trapping”. Our proof relies on a recently developed theory that relates the convergence of processes to that of associated resistance metric measure spaces.
{"title":"Anomalous scaling regime for one-dimensional Mott variable-range hopping","authors":"David A. Croydon, Ryoki Fukushima, Stefan Junk","doi":"10.1214/22-aap1915","DOIUrl":"https://doi.org/10.1214/22-aap1915","url":null,"abstract":"We derive an anomalous, sub-diffusive scaling limit for a one-dimen-sional version of the Mott random walk. The limiting process can be viewed heuristically as a one-dimensional diffusion with an absolutely continuous speed measure and a discontinuous scale function, as given by a two-sided stable subordinator. Corresponding to intervals of low conductance in the discrete model, the discontinuities in the scale function act as barriers off which the limiting process reflects for some time before crossing. We also discuss how, by incorporating a Bouchaud trap model element into the setting, it is possible to combine this “blocking” mechanism with one of “trapping”. Our proof relies on a recently developed theory that relates the convergence of processes to that of associated resistance metric measure spaces.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119731","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 concerns with the hydrodynamic limit of the Kob–Andersen model, an interacting particle system that has been introduced by physicists in order to explain glassy behavior, and widely studied since. We will see that the density profile evolves in the hydrodynamic limit according to a nondegenerate hydrodynamic equation, and understand how the diffusion coefficient decays as density grows.
{"title":"Hydrodynamic limit for the Kob–Andersen model","authors":"Assaf Shapira","doi":"10.1214/22-aap1898","DOIUrl":"https://doi.org/10.1214/22-aap1898","url":null,"abstract":"This paper concerns with the hydrodynamic limit of the Kob–Andersen model, an interacting particle system that has been introduced by physicists in order to explain glassy behavior, and widely studied since. We will see that the density profile evolves in the hydrodynamic limit according to a nondegenerate hydrodynamic equation, and understand how the diffusion coefficient decays as density grows.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996628","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}
Consider the random Cayley graph of a finite Abelian group $G$ with respect to $k$ generators chosen uniformly at random, with $1 ll log k ll log |G|$. Draw a vertex $U sim operatorname{Unif}(G)$. We show that the graph distance $operatorname{dist}(mathsf{id},U)$ from the identity to $U$ concentrates at a particular value $M$, which is the minimal radius of a ball in $mathbb Z^k$ of cardinality at least $|G|$, under mild conditions. In other words, the distance from the identity for all but $o(|G|)$ of the elements of $G$ lies in the interval $[M - o(M), M + o(M)]$. In the regime $k gtrsim log |G|$, we show that the diameter of the graph is also asymptotically $M$. In the spirit of a conjecture of Aldous and Diaconis (1985), this $M$ depends only on $k$ and $|G|$, not on the algebraic structure of $G$. Write $d(G)$ for the minimal size of a generating subset of $G$. We prove that the order of the spectral gap is $|G|^{-2/k}$ when $k - d(G) asymp k$ and $|G|$ lies in a density-$1$ subset of $mathbb N$ or when $k - 2 d(G) asymp k$. This extends, for Abelian groups, a celebrated result of Alon and Roichman (1994). The aforementioned results all hold with high probability over the random Cayley graph.
{"title":"Geometry of random Cayley graphs of Abelian groups","authors":"Jonathan Hermon, Sam Olesker-Taylor","doi":"10.1214/22-aap1899","DOIUrl":"https://doi.org/10.1214/22-aap1899","url":null,"abstract":"Consider the random Cayley graph of a finite Abelian group $G$ with respect to $k$ generators chosen uniformly at random, with $1 ll log k ll log |G|$. Draw a vertex $U sim operatorname{Unif}(G)$. We show that the graph distance $operatorname{dist}(mathsf{id},U)$ from the identity to $U$ concentrates at a particular value $M$, which is the minimal radius of a ball in $mathbb Z^k$ of cardinality at least $|G|$, under mild conditions. In other words, the distance from the identity for all but $o(|G|)$ of the elements of $G$ lies in the interval $[M - o(M), M + o(M)]$. In the regime $k gtrsim log |G|$, we show that the diameter of the graph is also asymptotically $M$. In the spirit of a conjecture of Aldous and Diaconis (1985), this $M$ depends only on $k$ and $|G|$, not on the algebraic structure of $G$. Write $d(G)$ for the minimal size of a generating subset of $G$. We prove that the order of the spectral gap is $|G|^{-2/k}$ when $k - d(G) asymp k$ and $|G|$ lies in a density-$1$ subset of $mathbb N$ or when $k - 2 d(G) asymp k$. This extends, for Abelian groups, a celebrated result of Alon and Roichman (1994). The aforementioned results all hold with high probability over the random Cayley graph.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996634","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}
It is well known that Glauber dynamics on spin systems typically suffer exponential slowdowns at low temperatures. This is due to the emergence of multiple metastable phases in the state space, separated by narrow bottlenecks that are hard for the dynamics to cross. It is a folklore belief that if the dynamics is initialized from an appropriate random mixture of ground states, one for each phase, then convergence to the Gibbs distribution should be much faster. However, such phenomena have largely evaded rigorous analysis, as most tools in the study of Markov chain mixing times are tailored to worst-case initializations. In this paper we develop a general framework towards establishing this conjectured behavior for the Ising model. In the classical setting of the Ising model on an N-vertex torus in Zd, our framework implies that the mixing time for the Glauber dynamics, initialized from a 12-12 mixture of the all-plus and all-minus configurations, is N1+o(1) in dimension d=2, and at most quasi-polynomial in all dimensions d≥3, at all temperatures below the critical one. The key innovation in our analysis is the introduction of the notion of “weak spatial mixing within a phase”, a low-temperature adaptation of the classical concept of weak spatial mixing. We show both that this new notion is strong enough to control the mixing time from the above random initialization (by relating it to the mixing time with plus boundary condition at O(logN) scales), and that it holds at all low temperatures in all dimensions. This framework naturally extends to more general families of graphs. To illustrate this, we use the same approach to establish optimal O(NlogN) mixing for the Ising Glauber dynamics on random regular graphs at sufficiently low temperatures, when initialized from the same random mixture.
{"title":"Low-temperature Ising dynamics with random initializations","authors":"Reza Gheissari, Alistair Sinclair","doi":"10.1214/22-aap1911","DOIUrl":"https://doi.org/10.1214/22-aap1911","url":null,"abstract":"It is well known that Glauber dynamics on spin systems typically suffer exponential slowdowns at low temperatures. This is due to the emergence of multiple metastable phases in the state space, separated by narrow bottlenecks that are hard for the dynamics to cross. It is a folklore belief that if the dynamics is initialized from an appropriate random mixture of ground states, one for each phase, then convergence to the Gibbs distribution should be much faster. However, such phenomena have largely evaded rigorous analysis, as most tools in the study of Markov chain mixing times are tailored to worst-case initializations. In this paper we develop a general framework towards establishing this conjectured behavior for the Ising model. In the classical setting of the Ising model on an N-vertex torus in Zd, our framework implies that the mixing time for the Glauber dynamics, initialized from a 12-12 mixture of the all-plus and all-minus configurations, is N1+o(1) in dimension d=2, and at most quasi-polynomial in all dimensions d≥3, at all temperatures below the critical one. The key innovation in our analysis is the introduction of the notion of “weak spatial mixing within a phase”, a low-temperature adaptation of the classical concept of weak spatial mixing. We show both that this new notion is strong enough to control the mixing time from the above random initialization (by relating it to the mixing time with plus boundary condition at O(logN) scales), and that it holds at all low temperatures in all dimensions. This framework naturally extends to more general families of graphs. To illustrate this, we use the same approach to establish optimal O(NlogN) mixing for the Ising Glauber dynamics on random regular graphs at sufficiently low temperatures, when initialized from the same random mixture.","PeriodicalId":50979,"journal":{"name":"Annals of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136167557","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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