Dynamics and Probability Seminar Spring 2022

The Dynamics and Probability seminar will meet on Tuesday afternoons at 2:30 in Cornett A120, or on Zoom when the speaker is not present in Victoria. All are welcome. Please contact Anthony Quas for the zoom link. The list of seminars is as follows. For further information, please contact me at aquas(a)uvic.ca or Gourab Ray at gourabray(a)uvic.ca

Date Speaker Title

18/1/2022 Matteo Tanzi (NYU) Random-like properties of chaotic forcing

25/1/2022 Paul Dario (Lyon) Random-field random surfaces

1/2/2022 Mark Pollicott (Warwick) Two geometric applications of Lyapunov exponents for random matrix products

8/2/2022 George Lee (UVic) TBA

15/2/2022 Vaughn Climenhaga (Houston) Counting closed geodesics

22/2/2022 Reading Break

1/3/2022 Sky Cao (Stanford) A state space for the 3D Yang-Mills measure

8/3/2022 Yinon Spinka (UBC) Entropy-efficient finitary codings by IID processes

15/3/2022 Matan Harel (Northeastern) On the relation between delocalization of integer-valued height functions and the Berezinskii-Kosterlitz-Thouless phase transition

22/3/2022 Rémy Mahfouf (ENS) Universality of spin correlations in the Ising model on isoradial graphs

29/3/2022 Zoe Huang (UBC) Co-evolving Dynamic Networks

5/4/2022 Alex Blumenthal (Georgia Tech) First steps towards a quantitative Furstenberg criterion and applications

Date: 18/1/2022
Speaker: Matteo Tanzi (NYU)
Title: Random-like properties of chaotic forcing
Abstract: We prove that skew systems with a sufficiently expanding base have "approximate" statistical properties similar to random ergodic Markov chains. For example, they exhibit approximate exponential decay of correlations, meaning that the exponential rate is observed modulo a controlled error. The fiber maps are only assumed to be Lipschitz regular and to depend on the base in a way that guarantees diffusive behaviour on the vertical component. The assumptions do not imply an hyperbolic picture and one cannot rely on the spectral properties of the transfer operators involved. The approximate nature of the result is the inevitable price one pays for having so mild assumptions on the dynamics on the vertical component. The error in the approximation is shown to go to zero when the expansion of the base tends to infinity.

Date: 25/1/2022
Speaker: Paul Dario (Lyon)
Title: Random-field random surfaces
Abstract: In 1975, the physicists Imry and Ma predicted that the incorporation of a random field in low-dimensional spin systems leads to the rounding of first-order phase transitions. These predictions were rigorously established by Aizenman and Wehr in 1989 for general spin systems, and recently quantified in the case of the random field Ising model. While the Imry-Ma phenomenon has been mainly studied in the case of compact spin spaces, it has been observed that a similar effect occurs for other models of statistical physics such as random surfaces. In this talk, we will present how the qualitative properties of these models are affected by the additon of a random field and discuss some open questions.

Date: 1/2/2022
Speaker: Mark Pollicott (Warwick)
Title: Two geometric applications of Lyapunov exponents for random matrix products
Abstract: Given square matrices A_1, ..., A_d we can consider random products and the associated (top) Lyapunov exponent. We will consider two applications where the Lyapunov exponent plays an interesting role: firstly to barycentric subdivisions of triangles in the Euclidean plane; and secondly to random walks in the hyperbolic plane.

Date: 8/2/2022
Speaker: George Lee (UVic)
Title:
Abstract:

Date: 15/2/2022
Speaker: Vaughn Climenhaga (Houston)
Title: Counting closed geodesics
Abstract: For negatively curved Riemannian manifolds, various natural geometric quantities grow exponentially quickly: the volume of a ball in the universal cover; the number of "distinguishable" geodesics of a given length; the number of closed geodesics with length below a given threshold. Margulis gave very precise asymptotic estimates in this setting. After surveying the general background and history of Margulis-type results, I will describe joint work with Gerhard Knieper and Khadim War in which we obtain Margulis asymptotics for surfaces without conjugate points.

Date: 1/3/2022
Speaker: Sky Cao (Stanford)
Title: A state space for the 3D Yang-Mills measure
Abstract: In this talk, I will describe some progress towards the construction of the 3D Yang-Mills (YM) measure. In particular, I will introduce a state space of “distributional gauge orbits” which may possibly support the 3D YM measure. Then, I will describe a result which says that assuming that 3D YM theories exhibit short distance behavior similar to the 3D Gaussian free field (which is the expected behavior), then the 3D YM measure may be constructed as a probability measure on the state space. The underlying technical details involve analyzing the YM heat flow (which is a certain PDE) with random distributional initial data. This is based on joint work with Sourav Chatterjee.

Date: 8/3/2022
Speaker: Yinon Spinka (UBC)
Title: Entropy-efficient finitary codings by IID processes
Abstract:A process Y is a factor of a process X if it can be written as Y=F(X) for some function F which commutes with translations. The factor is finitary if Y_0 is almost surely determined by some finite portion of the input X. Given a process Y, the question of whether Y is a (finitary) factor of an IID process is fundamental in ergodic theory and has received much attention in probability as well. As it turns out, contrary to the prevailing belief, some classical results about factors do not have finitary counterparts, as was recently shown by Gabor. We will present a complementary result that any process Y which is a finitary factor of an IID process furthermore admits an entropy-efficient finitary coding by an IID process. Here entropy-efficient means that the IID process has entropy arbitrarily close to that of Y. As an application we give an affirmative answer to an old question of van den Berg and Steif about the critical Ising model. Joint work with Tom Meyerovitch.

Date: 15/3/2022
Speaker: Matan Harel (Northeastern)
Title: On the relation between delocalization of integer-valued height functions and the Berezinskii-Kosterlitz-Thouless phase transition
Abstract: In this talk, we will discuss the relation between two types of two-dimensional lattice models: on one hand, we will consider the spin models with an O(2)-invariant interaction, such as the famous XY and Villain models. On the other, we study integer-valued height function models, where the interaction depends on the discrete gradient. We show that delocalization of a height function model implies that an associated O(2)-invariant spin model has a power-law decay phase. Motivated by this observation, we also extend the recent work of Lammers to show that a certain class of integer-valued height functions delocalize for all doubly periodic graphs (in particular, on the square lattice). Together, these results give a new perspective on the celebrated Berezinksii-Kosterlitz-Thouless phase transition for two-dimensional O(2)-invariant lattice models. This is joint work with Michael Aizenman, Ron Peled, and Jacob Shapiro.

Date: 22/3/2022
Speaker: Rémy Mahfouf (ENS)
Title: Universality of spin correlations in the Ising model on isoradial graphs
Abstract: We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs and Z–invariant weights. Specifically, we show that in the massive scaling limit, (i. e. as the mesh size tends to zero at the same rate as the inverse temperature goes to the critical one) the two-point spin correlations converges to a rotationally invariant function, which is universal among isoradial graphs and independant of the local geometry. We also give a simple proof of the fact that the infinite-volume sub-critical magnetization is independent of the site and the local geometry of the lattice. Finally, we provide a geometrical interpretation of the correlation length using the formalism of s-embeddings introduced recently by Chelkak. Based on a joint work (arXiv:2104.12858) with Dmitry Chelkak (ENS), Konstantin Izyurov (Helsinki).

Date: 29/3/2022
Speaker: Zoe Huang (UBC)
Title: Co-evolving Dynamic Networks
Abstract: Co-evolving network models, wherein dynamics such as random walks on the network influence the evolution of the network structure, which in turn influences the dynamics, are of interest in a range of domains. While much of the literature in this area is currently supported by numerics, providing evidence for fascinating conjectures and phase transitions, proving rigorous results has been quite challenging. We propose a general class of co-evolving tree network models driven by local exploration, started from a single vertex called the root. New vertices attach to the current network via randomly sampling a vertex and then exploring the graph for a random number of steps in the direction of the root, connecting to the terminal vertex. Specific choices of the exploration step distribution lead to the well-studied affine preferential attachment and uniform attachment models, as well as less well understood dynamic network models with global attachment functionals such as PageRank scores. We obtain local weak limits for such networks and use them to derive asymptotics for the limiting empirical degree and PageRank distribution. We also quantify asymptotics for the degree and PageRank of fixed vertices, including the root, and the height of the network. Two distinct regimes are seen to emerge, based on the expected exploration distance of incoming vertices, which we call the `fringe' and `non-fringe' regimes. These regimes are shown to exhibit different qualitative and quantitative properties. In particular, networks in the non-fringe regime undergo `condensation' where the root degree grows at the same rate as the network size. Networks in the fringe regime do not exhibit condensation. A non-trivial phase transition phenomenon is also displayed for the PageRank distribution, which connects to the well known power-law hypothesis. In the process, we develop a general set of techniques involving local limits, infinite-dimensional urn models, related multitype branching processes and corresponding Perron-Frobenius theory, branching random walks, and in particular relating tail exponents of various functionals to the scaling exponents of quasi-stationary distributions of associated random walks. These techniques are expected to shed light on a variety of other co-evolving network models.

Date: 5/4/2022
Speaker: Alex Blumenthal (Georgia Tech)
Title: First steps towards a quantitative Furstenberg criterion and applications
Abstract: I will present our recent results on estimating the Lyapunov exponents of weakly-damped, weakly-dissipated stochastic differential equations. Our primary tool is a new, mildly-quantitative version of Furstenberg’s criterion.

For (some) previous semesters, see