Percolation for Gaussian Processes
Speaker 
Lakshmi Priya M.E (Tel Aviv University, Israel)


When 
Oct 15, 2024
from 02:00 PM to 04:00 PM 
Where  LH111, First Floor 
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Simply put, a Gaussian process (GP) is a very nice model of a random function. In the talk and in what follows, we will restrict to smooth enough GPs f : R2 → R.
GPs are widely used to model several physical phenomena; for this it is important to understand their level sets. In their pioneering works, Nazarov–Sodin developed techniques to study the number of connected components of level sets {f + ` = 0}, where ` ∈ R. Complementary to this study is the question of percolation of level sets: does {f+` = 0} have an unbounded connected component? In recent years, several works have made significant progress towards answering this question.
In the talk, we will discuss one such contribution due to Muirhead–Rivera–Vanneuville (MRV): Let F : R2 → R be a C3smooth, centered, isotropic stationary Gaussian process (SGP) with some mild covariance decay. Then E` defined to be the event that {F + ` ≥ 0} has an unbounded connected component undergoes a phase transition at the critical level `crit = 0. That is, P(E`) = 0 when ` < 0, and P(E`) = 1 when ` > 0.
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The above result of MRV is reminiscent of the classical phase transition for Bernoulli percolation on Z2. The Gaussian process F lacks several nice properties which the Bernoulli percolation model has. Hence although the roadmap to proving the result of MRV is very similar to the analogous result for Bernoulli percolation, several steps involve new ideas inspired from works of Chatterjee and Tassion.
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In the talk, we will introduce Bernoulli percolation on Z2; define Gaussian processses; highlight the main steps in proving phase transition in the Bernoulli model, and the refine ments required for Gaussian processes.
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