Abstract: We study large random matrices with i.i.d. entries conditioned to have prescribed row and column sums (margins), a problem connected to matrix scaling, Schrödinger bridges, contingency tables, and random graphs with given degree sequences. Our central observation is a "transference principle": the complex margin-conditioned matrix can be closely approximated by a simpler matrix whose entries are independent and drawn from an exponential tilting of the original model. The tilt parameters are determined by the sum of two potentials, which can be computed by the generalized Sinkhorn algorithm at a dimension-free exponential rate. In fact, there is a hidden variational problem behind these seemingly unrelated problems: finding a matrix with given margins that minimizes a certain relative entropy. The potentials solve the so-called Kantorovich dual of this primal problem. The structure of the margin-conditioned random matrices is governed by the solution to this variational problem, which leads to, in particular, the sharp phase transition in contingency tables obtained by Dittmer, Lyu, and Pak in 2020. We establish phase diagrams for tame margins, where the potentials are uniformly bounded. As an application of this framework, we address a 2011 conjecture by Chatterjee, Diaconis, and Sly on δ-tame degree sequences for the β-model random graphs.