# Limit shapes for square ice and tails of the KPZ equation

Lead Research Organisation:
King's College London

Department Name: Mathematics

### Abstract

This research proposal is pointing at two fundamental problems in the theory of exactly solvable lattice models in statistical physics and the field of integrable probability, investigating various conjectures on scaling limits and universality behaviors for correlation and distribution functions. It aims at: (a) the exact description of limit shapes in the domain wall six-vertex model in its different phase regions; and (b) the derivation of tail expansions and large deviation principles for the Kardar-Parisi-Zhang equation. The central goal of this proposal is to discover a problem specific Riemann-Hilbert approach for both projects (a) and (b) and complete them through the development of novel nonlinear steepest descent techniques in combination with original ideas and techniques coming from random matrix theory and integrable systems.

The six-vertex model is the prototypical vertex integrable model for two-dimensional crystals with hydrogen bonding. It was introduced by Pauling as model for a flat H2O crystal and famously analyzed by Lieb and Sutherland for periodic boundary conditions. Subject to domain wall boundary conditions, the six-vertex model generalizes the dimer model on the Aztec diamond as well as ensembles of enumerated alternating sign matrices. Yet, the integrability in the six-vertex model with domain wall boundary conditions is fundamentally different from the determinantal or Pfaffian structures encountered in tiling or dimer models. In turn, almost nothing has been rigorously established about the six-vertex model's general geometry and its limit shapes. This fact identifies strand (a) as a central problem in mathematical statistical mechanics.

The celebrated Kardar-Parisi-Zhang (KPZ) equation has become the quintessential model for random surface growth processes with numerous remarkable connections to a number of different physical phenomena. Despite several impressive results in recent KPZ literature there is still a substantial lack of fine solution properties, for instance rigorous lower tail expansions are poorly understood. I propose to derive such estimates by developing a nonlinear steepest descent method for operator-valued Riemann-Hilbert problems. This is an analytical approach to a problem in integrable probability and stochastic analysis which was previously inaccessible from either field. This fact identifies strand (b) as a current important problem in integrable probability which will be solved through the development of novel integrable systems techniques. Thus, firmly placing one of the most celebrated stochastic PDEs in the realm of integrable systems.

The results of this proposal will resolve long-standing conjectures in statistical physics and integrable probability that have attracted considerable interest over the past 15 years but which were previously inaccessible by rigorous methods. Alongside the solution of strands (a) and (b), the proposed approach to both projects develops powerful mathematical techniques for the analysis of scaling and universality behaviors in mathematical physics and will thus have broad impact in other areas of mathematics and science. To be precise, I fully expect that the proposal's interdisciplinary character and mathematical results will impact the following physical problems: the theory of critical phenomena and phase separations, random growth models, combinatorial asymptotics in quantum gravity, lattice models in statistical physics, interacting particle systems, and others.

The six-vertex model is the prototypical vertex integrable model for two-dimensional crystals with hydrogen bonding. It was introduced by Pauling as model for a flat H2O crystal and famously analyzed by Lieb and Sutherland for periodic boundary conditions. Subject to domain wall boundary conditions, the six-vertex model generalizes the dimer model on the Aztec diamond as well as ensembles of enumerated alternating sign matrices. Yet, the integrability in the six-vertex model with domain wall boundary conditions is fundamentally different from the determinantal or Pfaffian structures encountered in tiling or dimer models. In turn, almost nothing has been rigorously established about the six-vertex model's general geometry and its limit shapes. This fact identifies strand (a) as a central problem in mathematical statistical mechanics.

The celebrated Kardar-Parisi-Zhang (KPZ) equation has become the quintessential model for random surface growth processes with numerous remarkable connections to a number of different physical phenomena. Despite several impressive results in recent KPZ literature there is still a substantial lack of fine solution properties, for instance rigorous lower tail expansions are poorly understood. I propose to derive such estimates by developing a nonlinear steepest descent method for operator-valued Riemann-Hilbert problems. This is an analytical approach to a problem in integrable probability and stochastic analysis which was previously inaccessible from either field. This fact identifies strand (b) as a current important problem in integrable probability which will be solved through the development of novel integrable systems techniques. Thus, firmly placing one of the most celebrated stochastic PDEs in the realm of integrable systems.

The results of this proposal will resolve long-standing conjectures in statistical physics and integrable probability that have attracted considerable interest over the past 15 years but which were previously inaccessible by rigorous methods. Alongside the solution of strands (a) and (b), the proposed approach to both projects develops powerful mathematical techniques for the analysis of scaling and universality behaviors in mathematical physics and will thus have broad impact in other areas of mathematics and science. To be precise, I fully expect that the proposal's interdisciplinary character and mathematical results will impact the following physical problems: the theory of critical phenomena and phase separations, random growth models, combinatorial asymptotics in quantum gravity, lattice models in statistical physics, interacting particle systems, and others.

### Planned Impact

The overarching impact goal of the proposal is to raise awareness of mathematical statistical mechanics and integrable probability among the current and next generation of mathematicians. Both areas have seen spectacular advances over the past 10 years with deep connections to probability, mathematical physics, representation theory and integrable systems. Yet, across the universities in Greater London, both areas are significantly underrepresented in the student body. Key focus points are thus Impact on people (hiring of a PDRA, mentoring of student theses in integrable probability/statistical mechanics and supervision of summer research projects at King's College London in the same subjects), Outreach (advertisement of integrable probability and statistical mechanics at London teacher conferences and taster days as well as King's mathematics school) and Dissemination and Communication (publication in leading mathematics and mathematical physics journals, delivery of seminar lectures across Greater London, participation in national and international conferences and organization of similar workshops and conferences in statistical mechanics and integrable probability).

## People |
## ORCID iD |

Thomas Joachim Bothner (Principal Investigator) |

### Related Projects

Project Reference | Relationship | Related To | Start | End | Award Value |
---|---|---|---|---|---|

EP/T013893/1 | 01/07/2020 | 27/07/2020 | £267,185 | ||

EP/T013893/2 | Transfer | EP/T013893/1 | 27/10/2020 | 26/09/2023 | £265,588 |