An official website of the United States government
Erdos Renyi Laws for Exponentially and Polynomially Mixing Dynamical Systems
Erdős-Rényi limit laws give the length scale of a time-window over which time-averages in Birkhoff sums have a non-trivial almost-sure limit. We establish Erdős-Rényi type limit laws for Hölder observables on dynamical systems modeled by Young Towers with exponential and polynomial tails. This extends earlier results on Erdős-Rényi limit laws to a broad class of dynamical systems with some degree of hyperbolicity.
more »
« less
- Award ID(s):
- 2009923
- PAR ID:
- 10427557
- Date Published:
- Journal Name:
- Proceedings of the American Mathematical Society
- Volume:
- 151
- Issue:
- 151
- ISSN:
- 0002-9939
- Page Range / eLocation ID:
- 3425-3430
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
A classical Borel–Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will happen almost surely. In this article, we propose an extension of Borel–Cantelli Lemma to characterize the multiple occurrence of events on the same time scale. Our results imply multiple Logarithm Laws for recurrence and hitting times, as well as Poisson Limit Laws for systems which are exponentially mixing of all orders. The applications include geodesic flows on compact negatively curved manifolds, geodesic excursions on finite volume hyperbolic manifolds, Diophantine approximations and extreme value theory for dynamical systems.more » « less
-
Chakrabarti, Amit; Swamy, Chaitanya (Ed.)We consider the problem of sampling from the ferromagnetic Potts and random-cluster models on a general family of random graphs via the Glauber dynamics for the random-cluster model. The random-cluster model is parametrized by an edge probability p ∈ (0,1) and a cluster weight q > 0. We establish that for every q ≥ 1, the random-cluster Glauber dynamics mixes in optimal Θ(nlog n) steps on n-vertex random graphs having a prescribed degree sequence with bounded average branching γ throughout the full high-temperature uniqueness regime p < p_u(q,γ). The family of random graph models we consider includes the Erdős-Rényi random graph G(n,γ/n), and so we provide the first polynomial-time sampling algorithm for the ferromagnetic Potts model on Erdős-Rényi random graphs for the full tree uniqueness regime. We accompany our results with mixing time lower bounds (exponential in the largest degree) for the Potts Glauber dynamics, in the same settings where our Θ(n log n) bounds for the random-cluster Glauber dynamics apply. This reveals a novel and significant computational advantage of random-cluster based algorithms for sampling from the Potts model at high temperatures.more » « less
-
Scaling limits and universality: Critical percolation on weighted graphs converging to an 𝐿³ graphonWe develop a general universality technique for establishing metric scaling limits of critical random discrete structures exhibiting mean-field behavior that requires four ingredients: (i) from the barely subcritical regime to the critical window, components merge approximately like the multiplicative coalescent, (ii) asymptotics of the susceptibility functions are the same as that of the Erdős-Rényi random graph, (iii) asymptotic negligibility of the maximal component size and the diameter in the barely subcritical regime, and (iv) macroscopic averaging of distances between vertices in the barely subcritical regime. As an application of the general universality theorem, we establish, under some regularity conditions, the critical percolation scaling limit of graphs that converge, in a suitable topology, to an graphon. In particular, we define a notion of the critical window in this setting. The assumption ensures that the model is in the Erdős-Rényi universality class and that the scaling limit is Brownian. Our results do not assume any specific functional form for the graphon. As a consequence of our results on graphons, we obtain the metric scaling limit for Aldous-Pittel’s RGIV model inside the critical window (see D.J. Aldous and B. Pittel [Random Structures Algorithms 17 (2000), pp. 79–102]). Our universality principle has applications in a number of other problems including in the study of noise sensitivity of critical random graphs (see E. Lubetzky and Y. Peled [Israel J. Math. 252 (2022), pp. 187–214]). In Bhamidi et al. [Scaling limits and universality II: geometry of maximal components in dynamic random graph models in the critical regime, In preparation], we use our universality theorem to establish the metric scaling limit of critical bounded size rules. Our method should yield the critical metric scaling limit of Ruciński and Wormald’s random graph process with degree restrictions provided an additional technical condition about the barely subcritical behavior of this model can be proved (see A. Ruciński and N. C. Wormald [Combin. Probab. Comput. 1 (1992), pp. 169–180]).more » « less
-
Abstract Consider an ergodic measure preserving dynamical system ( T , X , μ ), and an observable ϕ : X → R . For the time series X n ( x ) = ϕ ( T n ( x )), we establish limit laws for the maximum process M n = max k ⩽ n X k in the case where ϕ is an observable maximized on a line segment, and ( T , X , μ ) is a hyperbolic dynamical system. Such observables arise naturally in weather and climate applications. We consider the extreme value laws and extremal indices for these observables on hyperbolic toral automorphisms, Sinai dispersing billiards and coupled expanding maps. In particular we obtain clustering and nontrivial extremal indices due to self intersection of submanifolds under iteration by the dynamics, not arising from any periodicity.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- The DOI is not currently available.
