MARK KAC SEMINAR

We investigate the concept of noise sensitivity for functionals of independent random variables, which refers to the property that a small perturbation in the underlying randomness leads to an asymptotically independent functional. We extend classical noise sensitivity criteria beyond the Boolean setting, deriving quantitative estimates with optimal rates.

As an application, we consider the model of directed polymers in random environments, which describes a random walk interacting with a random medium. In the critical dimension d=2 and under a critical rescaling of the noise strength, the partition function of the model is known to converge to a universal limit, the Stochastic Heat Flow. We show that in this regime, the partition function exhibits noise sensitivity.

(Based on joint work with Anna Donadini)

A random point in a d-dimensional unit ball is said to have a beta distribution if its density is proportional to $(1-|x|^2)^{\beta}$, where $\beta>-1$ is a parameter. For $\beta=0$ we recover the uniform distribution on the unit ball, the limiting case $\beta\to-1$ corresponds to the uniform distribution on the unit sphere, while the case $\beta \to\infty$ corresponds to the standard Gaussian distribution. Let $X_1,..., X_n$ be independent random points in the $d$-dimensional unit ball such that $X_i$ follows a beta distribution with parameter $\beta_i$. Their convex hull $[X_1,...,X_n]$ is called a beta polytope (with parameters $n, d, \beta_1,...,\beta_n$). We shall review results on the expected number of $k$-dimensional faces, expected volume (and other geometric functionals) of beta polytopes and two closely related classes of polytopes called beta' and the beta* polytopes. Several objects in stochastic geometry such as the typical cell of the Poisson-Voronoi tessellation or the zero cell of the homogeneous Poisson hyperplane tessellation (in Euclidean space or on the sphere) are related to beta' polytopes, while their analogues in the hyperbolic space are related to beta* polytopes. This allows for explicit computations for these objects.