Loop models on random planar maps
The aim of this course is to give an introduction to the statistical models on random planar maps which can be mapped to ensembles of loops and lines on the planar map, such as the O(n) loop model as well as the models enjoying sl(2) quantum group symmetry: the height RSOS and A-D-E models and the 6-vertex model.
The loop models on planar maps can be studied by analytic combinatorics or by being reformulated as large N matrix models. Both methods of solution are spelt out. The combinatorial approach is based on tessellating the random map with simpler elements which do not contain nested loop configurations. Concerning the matrix model techniques, I will sometimes refer to the previous course "Large-N Matrix models and Spectral Curves".
The solution is formulated in terms of a spectral curve, in general nonalgebraic. The singular points of the spectral curve describe universal scaling limits of random planar map models. The nature of the singularities reflects the properties of large weighted maps and can be understood in the framework of Liouville quantum gravity. In this sense the loop models on random planar maps represent discretisation of two-dimensional quantum gravity with continuous spectrum of the Virasoro central charge for the matter field.
The loop models on planar maps can be studied by analytic combinatorics or by being reformulated as large N matrix models. Both methods of solution are spelt out. The combinatorial approach is based on tessellating the random map with simpler elements which do not contain nested loop configurations. Concerning the matrix model techniques, I will sometimes refer to the previous course "Large-N Matrix models and Spectral Curves".
The solution is formulated in terms of a spectral curve, in general nonalgebraic. The singular points of the spectral curve describe universal scaling limits of random planar map models. The nature of the singularities reflects the properties of large weighted maps and can be understood in the framework of Liouville quantum gravity. In this sense the loop models on random planar maps represent discretisation of two-dimensional quantum gravity with continuous spectrum of the Virasoro central charge for the matter field.
讲师
Ivan Kostov
日期
2024年10月10日 至 12月05日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周二,周四 | 13:30 - 15:05 | A3-3-201 | ZOOM 12 | 815 762 8413 | BIMSA |
修课要求
Graduate level knowledge linear algebra and complex analysis. Basic knowledge of statistical mechanics.
课程大纲
1. The O(n) loop model on random maps. Maps and loop configurations. Statistical weights and partition functions. Phase diagram and critical points. Combinatorial solution: nested loop approach, loop equations, transfer matrix, functional equations. O(n) matrix model. Spectral curve, critical behaviour and
relation to Liouville gravity.
2. SOS, RSOS and A-D-E models on planar graphs. Statistical weights and partition functions. Formulation in terms of loop gas. Combinatorial solution by nested loop approach. Spectral curve, critical behaviour and relation to Liouville gravity. The A-D-E matrix models as minimal models of 2D quantum gravity. The SOS model on random maps and loop ensembles associated with affine Lie algebras of A-D-E type.
3. The six-vertex model on random maps. Statistical weights and partition functions. Reformulation as a loop model and combinatorial solution. The six-vertex matrix model. Critical behaviour and relation to Matrix Quantum Mechanics.
relation to Liouville gravity.
2. SOS, RSOS and A-D-E models on planar graphs. Statistical weights and partition functions. Formulation in terms of loop gas. Combinatorial solution by nested loop approach. Spectral curve, critical behaviour and relation to Liouville gravity. The A-D-E matrix models as minimal models of 2D quantum gravity. The SOS model on random maps and loop ensembles associated with affine Lie algebras of A-D-E type.
3. The six-vertex model on random maps. Statistical weights and partition functions. Reformulation as a loop model and combinatorial solution. The six-vertex matrix model. Critical behaviour and relation to Matrix Quantum Mechanics.
参考资料
[1] M. Gaudin and I. Kostov, O(n) model on a fluctuating random lattice: some exact results. Phys. Lett., B220:200, 1989.
[2] I. Kostov. Strings with discrete target space. Nucl. Phys., B376:539–598, 1992, http://arxiv.org/abshep-th/9112059.
[3] I. Kostov, Strings with discrete target space, Nucl.Phys. B376, 539–598(1992), arXiv:hep-th/9112059, http://arxiv.org/abs/hep-th/9112059
[4] I. Kostov, Exact solution of the six-vertex model on a random lattice, Nucl. Phys., B575:513–534, 2000, http://arxiv.org/abs/hep-th/9911023.
[5] I. Kostov, B. Ponsot, and D. Serban, Boundary liouville theory and 2d quantum gravity. Nucl.Phys., B683:309–362, 2004, http://arxiv.org/abs/hep-th/0307189
[6] I. K. Kostov, Boundary Loop Models and 2D Quantum Gravity, Lecture notes of the summer school on Exact methods in low-dimensional statistical physics and quantum computing, Les Houches, June 30 – August 1, 2008, Oxford Univ. Press, 2008.
[7] G. Borot, B. Eynard, Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies, J. Stat. Mech. 2011, P01010 (2011), arXiv:0910.5896
[8] [1] A. Elvey Price and P. Zinn-Justin. The six-vertex model on random planar maps revisited. Journal of Combinatorial Theory, Series A, 196:105739, 2023, arXiv:2007.07928
[9] G. Borot, J. Bouttier, and B. Duplantier, Nesting statistics in the O(n) loop model on random planar maps, Communications in Mathematical Physics, 404(3):1125–1229, 2023
[2] I. Kostov. Strings with discrete target space. Nucl. Phys., B376:539–598, 1992, http://arxiv.org/abshep-th/9112059.
[3] I. Kostov, Strings with discrete target space, Nucl.Phys. B376, 539–598(1992), arXiv:hep-th/9112059, http://arxiv.org/abs/hep-th/9112059
[4] I. Kostov, Exact solution of the six-vertex model on a random lattice, Nucl. Phys., B575:513–534, 2000, http://arxiv.org/abs/hep-th/9911023.
[5] I. Kostov, B. Ponsot, and D. Serban, Boundary liouville theory and 2d quantum gravity. Nucl.Phys., B683:309–362, 2004, http://arxiv.org/abs/hep-th/0307189
[6] I. K. Kostov, Boundary Loop Models and 2D Quantum Gravity, Lecture notes of the summer school on Exact methods in low-dimensional statistical physics and quantum computing, Les Houches, June 30 – August 1, 2008, Oxford Univ. Press, 2008.
[7] G. Borot, B. Eynard, Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies, J. Stat. Mech. 2011, P01010 (2011), arXiv:0910.5896
[8] [1] A. Elvey Price and P. Zinn-Justin. The six-vertex model on random planar maps revisited. Journal of Combinatorial Theory, Series A, 196:105739, 2023, arXiv:2007.07928
[9] G. Borot, J. Bouttier, and B. Duplantier, Nesting statistics in the O(n) loop model on random planar maps, Communications in Mathematical Physics, 404(3):1125–1229, 2023
听众
Graduate
, 博士后
, Researcher
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Ivan Kostov obtained his PhD in 1982 from the Moscow State University, with scientific advisers Vladimir Feinberg and Alexander Migdal. Then he worked in the group of Ivan Todorov at the INRNE Sofia, and since 1990 as a CNRS researcher at the IPhT, CEA-Saclay, France. Currently he is emeritus DR CNRS at IPhT and a visiting professor at UFES, Vitoria, Brazil.