Sohom Bhattacharya

The paper studies multilinear Gibbs measures whose Hamiltonian is a generalized U-statistic with a general base measure; under cut-norm convergence of the coupling matrices it expresses the limiting free energy as an infinite-dimensional optimization over functions, gives sufficient conditions for replica symmetry (constant optimizers) and uses counterexamples to show their necessity, and derives weak limits for local fields, the Hamiltonian and global magnetization, a universal weak law for contrasts n^-1Σc_iX_i→0 when Σc_i=o(n), exponential concentration bounds for local and global magnetizations, and existence of a sharp phase transition in the temperature parameter for higher-order interactions.

The paper studies multilinear Gibbs measures whose Hamiltonian is a generalized U-statistic with a general base measure; under cut-norm convergence of the coupling matrices it expresses the limiting free energy as an infinite-dimensional optimization over functions, gives sufficient conditions for replica symmetry (constant optimizers) and uses counterexamples to show their necessity, and derives weak limits for local fields, the Hamiltonian and global magnetization, a universal weak law for contrasts n^-1Σc_iX_i→0 when Σc_i=o(n), exponential concentration bounds for local and global magnetizations, and existence of a sharp phase transition in the temperature parameter for higher-order interactions.