Filippo COLOMO

Research project

Limit shape phenomenon in exactly solvable models of statistical mechanics

My current research mainly concerns the limit shape phenomenon in exactly solvable models of statistical mechanics. While the phenomenon has been investigated in depth in ‘free-fermionic’ models (dimers, tilings, non-intersecting lattice paths, etc.), with a wealth of spectacular developments, a lot remains to be understood in the ‘interacting’ case. An excellent training ground in this respect is the six-vertex model, which, although genuinely interacting, is nevertheless still exactly solvable.
My main achievements in this context are: (1) the analytic expression of the Arctic curve of the six-vertex model with domain-wall boundary conditions, for arbitrary value of its parameters; (2) as a spin-off of the above result, the analytic expression of the limit shape of large Alternating Sign Matrices; (3) the construction of the ‘tangent method’, allowing for the derivation of the Artic curve in a very wide class of models; (4) the study of the free-energy for the domino tilings of an Aztec diamond with a cut-off corner (or L-shaped domain), as a function of the aspect ratio of the region; we showed the occurrence of a third-order phase transition when the cut-off square, increasing in size, reaches the Arctic Circle, that is the separation curve of the original (unmodified) Aztec diamond.
Being a physicist, I sometimes lack the rigour required in mathematics. In this respect, it is fair to say that the above mentioned results are partially based on some unproven assumptions, and are thus still conjectural (albeit strongly supported). However, the method of (3), in applicaton to (2), has recently been proven in full rigour [A. Aggarwal, Invent. Math. 220 (2020) 611].
The questions I am presently addressing concern: i) a rigorous derivation of (1); ii) the derivation of some multiple integral representation for the one-point function of the six-vertex model with domain-wall boundary condition, and the evaluation of its asymptotic behaviour in the scaling limit; iii) the investigation of the fluctuations of the limit shape of the six-vertex model in vicinity of the Arctic Curve, where universal KPZ behaviour is expected; iv) in close relation with iii), the problem of the edge fluctuations of magnetization in quantum quenches of the XXZ quantum spin chain.
The technical tool-box in my investigations include integrability, with Algebraic Bethe Ansatz and the Quantum Inverse Scattering Method, and random matrix models, in particular discrete log-gases.
These scientific interests are shared with Luigi Cantini, at LPTM, Cergy Paris Université; we have an ongoing collaboration, and we hope to progress further on the above mentioned questions.