Norms of structured random matrices

Let us consider the structured m x n Gaussian matrix G_A=(a_{ij}g_{ij}), where g_{ij}'s are independent standard Gaussian variables and A=(a_{ij}) is a deterministic matrix. The exact behaviour of the spectral norm of the structured Gaussian matrix is known due to the result of Latała, van Handel, and Youssef from 2018. During the talk we will be interested in finding two-sided bounds for the expected value of the norm of G_A treated as an operator from l_p^n to l_q^m. We shall conjecture the sharp estimates expressed only in the terms of the coefficients a_{ij}'s. The conjectured lower bounds are true up to the constant depending only on p and q. We shall show that the upper bounds are true up to the multiplicative constant depending linearly on a certain (small) power of ln(mn). This is joint work with Radosław Adamczak, Joscha Prochno, and Michał Strzelecki.