An application of Talagrand’s inequality to the Linear Sigma Model

The Linear Sigma Model is the N-component and O(N)-invariant generalization of the well-known ${\mathrm{\Phi }}_2^4$ model. In the present work, we show that on $T^2$ at large N, each marginal distribution is close to a massive Gaussian free field, quantified in the 2-Wasserstein distance. The proof is a simple application of classical tools in Euclidean field theory combined with the Feyel/Üstünel variant of Talagrand's inequality. In contrast to prior work using stochastic quantization, our proof avoids perturbative assumptions on the mass or the coupling constant. Based on joint work with Matías Delgadino.