Local $h^*$-polynomials for one-row Hermite normal form simplices

The local $h^*$-polynomial of a lattice polytope is an important invariant arising in Ehrhart theory. Our focus in this work is on lattice simplices presented in Hermite normal form with a single non-trivial row. We prove that when the off-diagonal entries are fixed, the distribution of coefficients for the local $h^*$-polynomial of these simplices has a limit as the normalized volume goes to infinity. Further, this limiting distribution is determined by the coefficients for a relatively small normalized volume. We also provide a thorough analysis of two specific families of such simplices, to illustrate and motivate our main result.