Abstract: 

This talk will focus on the $O(N)$ Linear Sigma Model on $\mathbb{R}^{2}$ under a scaling dictated by the formal $1/N$ expansion.  We show that in the large $N$ limit, correlations decay exponentially fast, where the acquired mass decays exponentially in the inverse temperature.  In fact, each marginal converges to a massive Gaussian Free Field (GFF) on $\mathbb{R}^{2}$, quantified in the $2$-Wasserstein distance with a weighted $H^{1}(\mathbb{R}^{2})$ cost function.  In contrast to prior work on the torus via parabolic stochastic quantization, our results hold without restrictions on the coupling constants, allowing us to also obtain a massive GFF in a suitable double scaling limit.  Our proof combines the Feyel/\"Ust\"unel extension of Talagrand's inequality with some classical tools in Euclidean Quantum Field Theory.  Based on joint work with Matias Delgadino.