# Analysis Seminar

#### A Two-Sided Estimate for the Gaussian Noise Stability Deficit

Speaker: Ronen Eldan

Location: Warren Weaver Hall 1302

Date: Thursday, November 21, 2013, 11 a.m.

Synopsis:

The Gaussian Noise Stability of a set $$A \subset \mathbb{R}^n$$ with parameter $$0 <\rho < 1$$ is defined as $$S_\rho(A) = \mathbb{P}(X,Y \in A)$$ where $$X,Y$$ are jointly Gaussian random vectors such that $$X$$ and $$Y$$ are standard Gaussian vectors and $$\mathbb{E} [ X_i Y_j ] = \delta_{ij} \rho$$. Borell's celebrated noise stability inequality states that if $$H$$ is a half-space whose Gaussian measure is equal to that of $$A$$, then $$S_\rho(H) \geq S_\rho (A)$$ for all $$0 < \rho < 1$$.

We will present a novel short proof of this inequality, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality: by introducing a new metric to measure the distance between the set $$A$$ and its corresponding half-space $$H$$ (namely the distance between the two centroids), we show that the deficit between the noise stability of $$A$$ and $$H$$ can be controlled from both below and above by essentially the same function of the distance, up to logarithmic factors.

As a consequence, we also manage to get the conjectured exponent in the robustness estimate proven by Mossel-Neeman, which uses the total-variationdistance as a metric. Moreover, in the limit $$\rho \to 1$$, we get an improved dimension free robustness bound for the Gaussian isoperimetricinequality.