Project Details
Abstract
1) Wider research context
This project concerns lattice models, a major topic of research in probability, in particular since the Fields medals awarded to Smirnov and Werner. These models were introduced starting from the 1920s in relation to phase transition phenomena. In two dimensions, the development of renormalisation group and conformal field theories in physics led to a description of the expected limiting correlations at criticality. In 2000, Schramm revolutionised the area by adapting Loewner’s equation from conformal geometry to give a precise description of conjectural limits of interfaces in lattice models, via random fractal curves.
2) Objectives
We suggest a new approach to conformal invariance at an infinite-order transition of BKT type, named after the seminal work of Berezinskii–Kosterlitz–Thouless. More generally we aim to establish key aspects of phase transitions in two dimensions:
- first-order transitions with symmetry-breaking: convergence to Brownian bridge (random-cluster model);
- higher-order transitions: fractality of scaling limits at a surface of critical points and critical curve splitting in two (dilute Potts and Ashkin-Teller models)
- infinite-order transition: complete diagram (loop O(2) model).
3) Approach
Capitalising on recent developments from the PI and coauthors which reveal an underlying Fortuin-Kasteleyn-Ginibre (FKG) structure for certain models, we plan:
- to derive the renewal structure of an interface at first-order transitions and extend Ornstein-Zernike theory to non-symmetric phases;
- to establish monotonicity for the loop O(n) model and extend fractal behaviour from the tricritical line to the critical surface of parameters;
- to prove sharpness of the BKT transition in the loop O(2) model via transfer-matrix methods.
At the BKT transition, the idea is to establish exact solvability of the loop O(2) model by constructing its self-dual representation. This would yield new discrete-holomorphic observables and give links to the Gaussian Free Field, via the dimer model.
4) Innovation
The above results would solve some of the most exciting open problems in the field, and open new directions. Indeed so far there has been almost no rigorous result on infinite-order transitions, and conformal invariance has only been proved in a handful of cases. Even in first-order transitions, Ornstein-Zernike theory currently only applies to situations with enough symmetry, whereas our approach goes much beyond this in
two dimensions, obtaining the renewal structure directly from positive association. In higher-order transitions, proving monotonicity for the loop O(n) model would suggest a way to obtain scaling relations on universal critical exponents.
5) Primary researchers involved:
PI - Alexander Glazman
PhD student and Postdoc will be hired
International collaborators: de Tilière, Duminil-Copin, Laslier (all - Paris), Manolescu (Fribourg), Ott (Rome), Spinka (Vancouver)
This project concerns lattice models, a major topic of research in probability, in particular since the Fields medals awarded to Smirnov and Werner. These models were introduced starting from the 1920s in relation to phase transition phenomena. In two dimensions, the development of renormalisation group and conformal field theories in physics led to a description of the expected limiting correlations at criticality. In 2000, Schramm revolutionised the area by adapting Loewner’s equation from conformal geometry to give a precise description of conjectural limits of interfaces in lattice models, via random fractal curves.
2) Objectives
We suggest a new approach to conformal invariance at an infinite-order transition of BKT type, named after the seminal work of Berezinskii–Kosterlitz–Thouless. More generally we aim to establish key aspects of phase transitions in two dimensions:
- first-order transitions with symmetry-breaking: convergence to Brownian bridge (random-cluster model);
- higher-order transitions: fractality of scaling limits at a surface of critical points and critical curve splitting in two (dilute Potts and Ashkin-Teller models)
- infinite-order transition: complete diagram (loop O(2) model).
3) Approach
Capitalising on recent developments from the PI and coauthors which reveal an underlying Fortuin-Kasteleyn-Ginibre (FKG) structure for certain models, we plan:
- to derive the renewal structure of an interface at first-order transitions and extend Ornstein-Zernike theory to non-symmetric phases;
- to establish monotonicity for the loop O(n) model and extend fractal behaviour from the tricritical line to the critical surface of parameters;
- to prove sharpness of the BKT transition in the loop O(2) model via transfer-matrix methods.
At the BKT transition, the idea is to establish exact solvability of the loop O(2) model by constructing its self-dual representation. This would yield new discrete-holomorphic observables and give links to the Gaussian Free Field, via the dimer model.
4) Innovation
The above results would solve some of the most exciting open problems in the field, and open new directions. Indeed so far there has been almost no rigorous result on infinite-order transitions, and conformal invariance has only been proved in a handful of cases. Even in first-order transitions, Ornstein-Zernike theory currently only applies to situations with enough symmetry, whereas our approach goes much beyond this in
two dimensions, obtaining the renewal structure directly from positive association. In higher-order transitions, proving monotonicity for the loop O(n) model would suggest a way to obtain scaling relations on universal critical exponents.
5) Primary researchers involved:
PI - Alexander Glazman
PhD student and Postdoc will be hired
International collaborators: de Tilière, Duminil-Copin, Laslier (all - Paris), Manolescu (Fribourg), Ott (Rome), Spinka (Vancouver)
| Status | Active |
|---|---|
| Effective start/end date | 1/08/21 → 31/07/25 |
Keywords
- phase transition
- conformal invariance
- positive correlation inequalities
- Ornstein-Zernike theory
- percolation
- Russo-Seymour-Welsh theory