Six-vertex model and random matrix distributions

We survey the connections between the six-vertex (square ice) model of 2D statistical mechanics and random matrix theory. We highlight the same universal probability distributions appearing on both sides, and also indicate related open questions and conjectures. We present full proofs of two asymptotic theorems for the six-vertex model: in the first one the Gaussian Unitary Ensemble (GUE) and GUE–corners process appear; the second one leads to the Tracy–Widom distribution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 2"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">F_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . While both results are not new, we found shorter transparent proofs for this text. On our way we introduce the key tools in the study of the six-vertex model, including the Yang–Baxter equation and the Izergin–Korepin formula.