Theoretical Statistics and Mathematics Division
Indian Statistical Institute, Kolkata.
203 Barrackpore Trunk Road, Kolkata 700108, IndiaAshok Maitra Memorial Lectures 2023-24
Prof. Maria Eulalia-Vares
Instituto de Matemática, Brazil.http://www.im.ufrj.br/eulalia//
will deliver the Ashok Maitra Memorial Lectures 2023-24 at
- Indian Statistical Institute, Bangalore Centre (February 15, 2024; 2 - 3 PM and 3.15 - 4.15 PM)
- Indian Statistical Institute, Kolkata Centre (February 21 and 22, 2024 ; 4.15-5.15 PM)
- Indian Statistical Institute, Delhi Centre (February 26 and 28, 2024; 3.30-4.30 PM)
At Indian Statistical Institute, Bangalore Centre
Lecture 1Title: Ising and percolation under highly anisotropic scaling limits
Date: February 15th, 2024 Time: 2-3 PM
Venue: Auditorium (II floor)
Youtube Link : https://www.youtube.com/watch?v=Y9IKzLoC2jQ
Abstract: In this lecture, I plan to motivate the consideration of a class of ferromagnetic Ising spin models with two types of interaction. Restricting ourselves to two dimensional systems, and setting the spins on a same horizontal line (layer) to interact through a Kac type potential, while vertical interactions are taken as nearest neighbors. This kind of interaction appears quite naturally and was indeed already mentioned by Kac. Nevertheless not much is known about it, as the general theory known as Lebowitz-Penrose approach does not apply when the Kac interaction is restricted to subspaces of positive co-dimension. The ultimate goal could be thought of as a description of the phase diagram for such models. In this lecture I will discuss the basic ideas and report on some (rather initial) progress that deals with horizontal interactions at the critical mean field value. Some open questions and conjectures are mentioned/discussed
The talk is based on the paper [2] below.
Lecture 2
Title: Ising and percolation under highly anisotropic scaling limits
Date: February 15th, 2024 Time: 3.15-4.15 PM
Venue: Auditorium (II floor)
Youtube Link : https://www.youtube.com/watch?v=Y9IKzLoC2jQ
Abstract: When restricted to the Bernoulli percolation set up, the discussion of the previous lecture at the critical mean field parameter brings to a question regarding the existence of a critical exponent for the opening probability of the vertical bonds that allows to see a phase transition. More precisely, one considers Bernoulli percolation on the anisotropic graph on the two dimensional lattice Z^2 which sets horizontal edges for each pair of vertices within a given Euclidean distance N and vertical edges connect only nearest neighbor vertices. Horizontal edges are open with probability 1/(2N), which corresponds to the critical mean field value, and one seeks for the critical exponent for the opening probability of the vertical bonds which would allow us to observe a phase transition for all large N. This brings us naturally to an analysis of the scaling limit of the growth process restricted to each horizontal layer, inspired by the work of Mueller and Tribe on the long range contact process. A renormalization scheme is used for the percolative regime.
This talk is based on the paper [3] below.
References:
[1] Cassandro, M.; Colangeli, M.; Presutti, E. Highly anisotropic scaling limits. Jr. Stat. Phys, 162, 997-1030, 2016. DOI 10.1007/s10955-015-1437-0
[2] Fontes, L.R. ; Marchetti, D. H. U. ; Merola, I.; Presutti, E.; Vares. Layered Systems at the Mean Field Critical Temperature. Jr. Stat. Phys. 161, 91-122, 2015.
[3] Mountford, T.S.; Vares, M.E.; Xue, H. Critical scaling for an anisotropic percolation system on Z^2,Elect. Jr. Probab. 25, p. 129, 44pp. 2020.
At Indian Statistical Institute, Kolkata
Public LectureTitle: Metastability for a class of stochastic dynamics
Date: February 21, 2024 Time: 4.15 - 5.15 PM
Venue:L-infinity (5th floor, Kolmogorov Bhavan)
Youtube Link : https://www.youtube.com/@csscisikolkata5034/streams
Abstract: Metastability is a ubiquitous phenomenon in nature, with examples coming from different fields, including physics, chemistry, biology, climatology and economics. The objects of interest are systems that make transitions between metastable states or quasi-equilibria and the stable state. Metastable behavior is characterized by a long period of apparent equilibrium of a pure thermodynamical phase followed by an unexpected fast decay towards the stable equilibrium of a pure phase or a mixture. Interesting questions in metastability involve: the transition time, the characterization of a possible critical droplet that precipitates the fast convergence to equilibrium. I plan to recall some general aspects of metastability, keeping a closer look into the frame of stochastic dynamics. Based on a few concrete examples, I hope to discuss some of the motivations and to describe general features, concluding with a soft description of results for the stochastic Ising model in the two-dimensional lattice at fixed subcritical temperature, as obtained in [1].
References:
[1] Gaudillière, A. ; Milanesi, P. ; Vares, M. E. Asymptotic Exponential Law for the Transition Time to Equilibrium of the Metastable Kinetic Ising Model with Vanishing Magnetic Field. Jr. Stat. Phys. 179, 263 - 308, 2020. https://doi.org/10.1007/s10955-019-02463-5
Two monographs on the subject:
[2] Bovier, A; den Hollander, F. Metastability. A Potential-Theoretic Approach. (Springer, 2015)
[3] Olivieri, E.; Vares, M.E. Large Deviations and Metastability. (Cambridge University Press, 2004)
Colloquium
Title:Critical scaling for anisotropic percolation system on Z^2
Date: February 22, 2024 Time: 4.15 - 5.15 PM
Venue:L-infinity (5th floor, Kolmogorov Bhavan)
Youtube Link : https://www.youtube.com/@csscisikolkata5034/streams
Abstract: Motivated by a question that arose in the frame of ferromagnetic Ising models with a layered interaction, I plan to discuss one further example of the interplay between SPDEs and percolation methods. For concreteness, we consider the following anisotropic graph on the two dimensional lattice Z^2: horizontal edges are set for each pair of vertices within a given Euclidean distance N and vertical edges connect only nearest neighbor vertices. On this graph one considers a Bernoulli percolation model, so that horizontal edges are open with probability 1/(2N), corresponding to the critical mean field value. The goal is the determination of a critical exponent for the opening probability of the vertical bonds which would allow us to observe a phase transition for all large N. This brings us naturally to an analysis of the scaling limit of the growth process restricted to each horizontal layer. This is inspired by the work of Mueller and Tribe on the long range contact process and plays a key role. A renormalization scheme is used for the percolative regime.
The talk is mostly based on a joint work with Thomas Mountford and Hao Xue [1].
Reference:
[1] T.S. Mountford, M. E. Vares, H. Xue. Critical scaling for an anisotropic percolation system on Z^2,Electronic Jr. Probab. 25, p. 129, 44pp. 2020
At Indian Statistical Institute, Delhi Centre
Lecture 1
Title: Renewal contact process and contact process on dynamic random environments
Date:Monday, February 26, 2024 Time:3:30 PM -4:30 PM
Venue:Room # 23 on First Floor of the Academic Block
Zoom Link :
Join Zoom Meetings
https://us02web.zoom.us/j/87004905511?pwd=MWtDQUZ2WFcwZUc4Y2JONTZWOEpjZz09
Meeting ID: 870 0490 5511
Passcode: AMML2324
Abstract: In this talk, I will first present recent progress on alternate versions of the Harris contact process, where some aspects of the dynamics (transmission or recovery) are governed by a general renewal process. Several of the tools employed in the study of the classical process no longer apply, and new ones are required. We present conditions on the inter-arrival distribution of the renewal process that ensure survival or extinction of the infection.
We then focus on a model introduced by Linker and Remenik (2020), namely the contact process evolving on a dynamical graph given by dynamic bond percolation on the edges of Z^d. The environment is thus defined in terms of two parameters corresponding to the percolation density p and the edge update rate v. Their work contains several interesting results about this model, including a result in dimension one that we extend to higher dimensions. It states that, for any fixed value of p below the critical threshold and the contact infection rate lambda, the infection dies out if v is small enough.
Lecture 2
Title: Renewal contact process and contact process on dynamic random environments
Date:Wednesday, February 28, 2024 Time:3:30 PM -4:30 PM
Venue:Room # 23 on First Floor of the Academic Block
Zoom Link :
Join Zoom Meetings
https://us02web.zoom.us/j/87004905511?pwd=MWtDQUZ2WFcwZUc4Y2JONTZWOEpjZz09
Meeting ID: 870 0490 5511
Passcode: AMML2324
Abstract: Continuing previous work on a class of dynamic environments for the contact process, we investigate extinction or survival of the infection when the agents perform an exclusion process on Z^d. This is an ongoing work in collaboration with M. Hilário, D. Ungaretti, and D. Valesin. In controlling the dependencies over the random environment a crucial role is played by the decoupling techniques obtained by R. Baldasso and A. Teixeira (2018).
References:
[1] Fontes, L.R; Mountford, T. S.; Ungaretti, D.; Vares, M.E. Renewal Contact Processes: Phase transition and survival. Stoch. Proc. Applic. 161, 102--136, 2023.
[2] Hilário, M.; Ungaretti, D.; Valesin, D.; Vares, M.E. Results on the contact process with dynamic edges or under renewals. Elect. Jr. Probab. 27, p. 91, 31 pp, 2022.
[3] Fontes, L.R.; Mountford, T.S.; Vares, M.E. Contact process under renewals II. Stoch. Proc. Applic. 130, 1103--1118, 2020.
Video recordings of some of the lectures will be uploaded here : https://www.youtube.com/playlist?list=PL1eY4X87dSYFea4TvtPkJr1L9wFV1NdDX
About Ashok Prasad Maitra
Past AMML lectures:
2022-2023: https://www.isibang.ac.in/~statmath/amml22-23/
2021-2022: https://www.isibang.ac.in/~statmath/amml22/
2020-2021: https://www.isibang.ac.in/~statmath/amml20/
2019-2020: https://www.isibang.ac.in/~statmath/amml19/
2011-2019: https://smu.isical.ac.in/menufour/ashok-maitra-memorial-lectures
Theoretical Statistics and Mathematics Unit
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Last modified :February 2024