The five-vertex model as a discrete log-gas

F. Colomo^a, M. Mannatzu^b, A. G. Pronko<sup>c

a</sup> INFN, Sezione di Firenze Via G. Sansone 1, 50019 Sesto Fiorentino (FI), Italy
^b Dipartimento di Fisica e Astronomia, Università di Firenze Via G. Sansone 1, 50019 Sesto Fiorentino (FI), Italy
^c St. Petersburg Department of V. A. Steklov Mathematical Institute, Fontanka 27, 191023 St. Petersburg, Russia

Abstract: We consider the five-vertex model on a rectangular domain of the square lattice, with the so-called “scalar-product” boundary conditions. We address the evaluation of the free-energy density of the model in the scaling limit, that is when the number of sites is sent to infinity and the mesh of the lattice to zero, while keeping the size of the domain constant. To this aim, we reformulate the partition function of the model in terms of a discrete log-gas, and study its behaviour in the thermodynamic limit. We reproduce previous results, obtained by using a differential equation approach. Moreover, we provide the explicit form of the resolvent in all possible regimes. This work is preliminary to further studies of limit shape phenomena in the model.

Key words and phrases: Scalar product boundary conditions, Hankel determinants, matrix models, third-order phase transition, plane partitions.

UDC: 531.19, 519.2

Received: 10.11.2025

Language: English