Scaling limit of Sinkhorn-rescaled Random Matrices via Stability of Static Schr\"odinger Bridges
We analyze the asymptotic behavior and scaling limits of large random matrices rescaled via the Sinkhorn algorithm to match prescribed row and column margins. For a random matrix with independent sub-exponential entries, we show that its Sinkhorn rescaling concentrates around the rescaling of its mean matrix, both at the level of the Schr\"odinger potentials and as random measures on the unit square, with explicit non-asymptotic rates. As the dimensions grow, the rescaled random matrix converges to the continuous static Schr\"odinger bridge (SSB) determined by the limiting margins and reference density. Around this scaling limit we develop a fluctuation theory: bulk rigidity for the empirical spectral distribution of the associated sample covariance matrix, and a central limit theorem for the empirical Schr\"odinger potentials of the rescaled empirical mean. Our analysis is driven by a new quantitative stability theory for the SSB, developed in three forms: Lipschitz continuity in the Hellinger distance under perturbations of the reference measure (kernel stability); H\"older-$1/2$ continuity in the Hellinger distance under $L^1$ perturbations of the margins (margin stability); and $L^\infty$ stability of the discrete Schr\"odinger potentials under margin perturbation (potential stability). Translated to the discrete random-matrix setting, these bounds yield the concentration and scaling-limit results, while a local law for random Gram matrices with a non-uniform variance profile drives the bulk rigidity. Our SSB stability theory may be of independent interest.