School on Random Geometry and Random Matrices

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Start time: August 25, 2014

Ends on: September 2, 2014

Location: São Paulo, Brazil

Venue: IFT-UNESP

Organizers:

Diego Trancanelli (USP), Stefan Zohren (PUC – Rio de Janeiro)

Lecturers:

  • Jérémie Bouttier (Saclay, France)
  • Zdzislaw Burda (Krakow, Poland)
  • François David (Saclay, France)
  • Nadav Drukker (King’s College, UK)
  • Thordur Jonsson (Iceland)

Description:

Models of random geometry and random matrices have wide applications, ranging from quantum gravity and string theory to complex networks and biological applications. In this interdisciplinary school, some of the world’s leading experts will give an overview of these diverse areas. The school is intended for graduate students and researchers in the fields of high energy physics, statistical physics and probability theory, and will be followed by a 2-day Workshop (September 3-4). The application for the school automatically includes participation in the workshop. There is no registration fee and limited funds are available for local and travel support of participants. This event received support from the Nordic Institute for Theoretical Physics (Nordita) for the participation of Nordic scientists.

 Application deadline: June 27

Announcement

Lectures Summary:

Jérémie Bouttier (Saclay)Recent developments in random planar maps, or the virtue of discreteness
The purpose of these lectures is to provide a gentle introduction to some recent developments in the theory of random planar maps. These objects form important models of random geometry, and appear as discretizations of 2D quantum gravity as well as in the topological expansion of matrix models.

We will focus on discrete methods: how to count maps (i.e. evaluate their partition function), how to study their geometric properties (mostly using bijections with trees), how to define a useful notion of thermodynamic limit (the so-called local limit). Due to time constraints we must leave aside other recent interesting developments regarding the continuum limits of maps (Brownian map and all that), maps of higher genera, the grand unification of bijections via orientations, maps with matter, etc.

Plan:

1) Introduction: definitions, motivations and basic properties of planar maps. Recursive decomposition.

2) Bijections: Schaeffer, BDG, Miermont, Ambjørn-Budd, etc.

3) Distance statistics: the two-point function, the three-point function, geodesics…

4) Local limits: UIPT/UIPQ, half-plane case, peeling process, application to the study of percolation on random maps

References:

  • J. Ambjørn, B. Durhuus and T. Jonsson, Quantum Geometry — A statistical field theory approach. Cambridge University Press, 1997.
  • J. Bouttier, P. Di Francesco and E. Guitter, Planar maps as labeled mobiles. Electron. J. Combin. 11 (2004) R69.
  • J. Ambjørn and T. Budd, Trees and spatial topology change in causal dynamical triangulations. J. Phys. A: Math. Theor. 46 (2013) 315201.
  • J. Bouttier and E. Guitter, Confluence of geodesic paths and separating loops in large planar quadrangulations. J. Stat. Mech. (2009) P03001.
  • O. Angel and O. Schramm, Uniform Infinite Planar Triangulations. Commun. Math. Phys. 241 (2003) 191–213.

Zdzislaw Burda (Krakow)Products of random matrices and their applications
The main objective of the course is to systematically develop a technique to calculate eigenvalue densities of products of invariant random matrices in the large N limit. The technique is based on enumeration of planar diagrams.

Outline:
1 Introduction to random matrices
2 Eigenvalue density and Green function
3 Planar Feynman diagrams
4 Extension to non-Hermitian matrices
5 Linearization – a trick to calculate Green functions for products of matrices
6 Relation of planar diagrams to free probability: R and S transforms
7 Some examples and applications

References:

  • E. Brezin, C. Itzykson, G. Parisi, J.-B. Zuber: Planar diagrams Commun. Math. Phys. 59, 35 (1978)
  • D. Bessis, C. Itzykson, J.-B. Zuber: Quantum field theory techniques in graphical enumeration. Adv. Appl. Math. 1, 109 (1980)
  • Z. Burda, R. A. Janik, B. Waclaw: Spectrum of the Product of Independent Random Gaussian Matrices. Phys. Rev. E 81, 041132 (2010)
  • Z.Burda, R.A. Janik, M.A. Nowak: Multiplication law and S transform for non-hermitian random matrices. Phys. Rev. E 84, 061125 (2011)
  • Z. Burda: Free products of large random matrices – a short review of recent developments. J. Phys.: Conf. Ser. 473, 012002 (2013)

François David (Saclay) - Liouville theory, KPZ and SLE
This course will be a short introduction, from a theoretical physicist’s point of view, of the relations between conformal field theories (CFT), critical statistical 2 dimensional systems, two dimensional quantum gravity and stochastic evolution process such as SLE. If time permits some recent developments will be outlined.

Plan of the course:

1 – A crash introduction to QFT and CFT, critical statistical systems and their relation with stochastic processes
2 – 2D quantum gravity and Liouville theory, the KPZ relations
3 – SLE processes and conformal invariance
4 – The KPZ relations from the probabilistic point of view, conformal welding, recent developments

A (very partial) introductory reference list:

  • P. Di Francesco, P. Mathieu, D. Senechal: Conformal Field Theory. Graduate Texts in Contemporary Physics, Springer.
  • See the contributions by M. Henkel and D. Karevski (CFT) and by M. Bauer (SLE and CFT): Conformal Invariance: an Introduction to Loops, Interfaces and Stochastic Loewner Evolution. Lecture Notes in Physics Volume 853 (2012); .
  • M. Bauer & D. Bernard: 2D growth processes: SLE and Loewner chains. Physics Reports 432 (2006) 115–221
  • A. Bovier, F. Dunlop, F. den Hollander, A. van Enter and J. Dalibard: Some recent aspects of random conformally invariant systems, W. Werner; in Les Houches, Session LXXXIII, 2005, Mathematical Statistical Physics. eds., pp. 101-217, Elsevier B. V. (2006)
  • A. Bovier, F. Dunlop, F. den Hollander, A. van Enter and J. Dalibard: Conformal Random Geometry, B. Duplantier, in Les Houches, Session LXXXIII, 2005, Mathematical Statistical Physics, , eds., pp. 101-217, Elsevier B. V. (2006)
  • See the contributions by J. Cardy, W. Werner, I. Kostov & B. Duplantier: Exact Methods in Low-dimensional Statistical Physics and Quantum Computing. Lecture Notes of the Les Houches Summer School: Volume 89, July 2008, Oxford University Press.

Nadav Drukker (King’s College)Matrix model for supersymmetric field theories
Field theories on continuous space have infinite numbers of degrees of freedom making the path integral very complicated (if at all well defined). Yet, it was realized in recent years that due to supersymmetry in certain very specific situations most of the modes get frozen and the dynamics can be described by a finite number of degrees of freedom. This reduces the full path integral to a finite dimensional one – a matrix model. In some examples these are well known models that have been solved before and in others these are more complicated models not considered previously. I will present the calculation of the partition function of supersymmetric 3d theories on S^3. I will explain the use of supersymmetric localization to reduce it to a zero dimensional matrix model. I will review different matrix model techniques for solving this matrix model both in the large N limit and to all orders in 1/N. I will also discuss the holographic dual of some of these models – string theory (or gravity) on 4d hyperbolic space (AdS_4) times a compact manifold (in the simplest case CP^3). In particular how to match the free energy of the matrix model with a gravitational calculation, both the large N limit (classical gravity) and the full genus expansion (quantum gravity).

References:

http://arxiv.org/abs/arXiv:1104.0783
http://arxiv.org/abs/arXiv:1110.4066
http://arxiv.org/abs/arXiv:1406.0505 

Thordur Jonsson (Iceland) – Random tree ensembles and applications
The Galton-Watson process and simply generated trees.  Properties of simply generated trees: the spine, condensation, Hausdorff and spectral dimensions.  Splitting vertex trees and some of their properties.

Some references and background are the following:

  • J. Ambjorn, B. Durhuus and T. Jonsson: Quantum geometry – A statistical field theory approach. Cambridge University Press, 1997.
  • B. Durhuus: Probabilistic aspects of infinite trees and surfaces. Acta Physica Polonica B (2003) 4795-4811.
  • B. Durhuus, T. Jonsson and J. Wheater: The spectral dimension of generic trees. J. Stat. Phys. 128 (2007) 1237-1260.
  • T. Jonsson and S. Ö. Stefánsson: Condensation in nongeneric trees. J. Stat. Phys. 142 (2011) 277-313.
  • F. David, W. M. B. Dukes, T. Jonsson and S. Ö. Stefansson: Random tree growth by vertex splitting. J. Stat. Mech. (2009) P04009

Programme: pdf programme_updated on August 21

FIRST WEEK: August 25 to 30

Monday, August 25

8:30  –  9:45 REGISTRATION
9:45 – 11:00 LECTURE I: T. Jonsson
11:00  – 11:30 COFFEE BREAK
11:30  – 12:45 LECTURE I: J. Bouttier
12:45 – 14:15 LUNCH
14:15 – 15:30 LECTURE I: Z. Burda
15:30 – 16:00 COFFEE BREAK
16:00 – 17:00 LECTURE I – Exercises: T. Jonsson
17:00 – 18:00 LECTURE I - Exercises / Solution: J. Bouttier
18:00 – 19:00 LECTURE I – Exercises: Z. Burda

Tuesday, August 26

9:45 – 11:00 LECTURE II: Z. Burda
11:00  – 11:30 COFFEE BREAK
11:30  – 12:45 LECTURE II: J. Bouttier
12:45 – 14:15 LUNCH
14:15 – 15:30 LECTURE II: T. Jonsson
15:30 – 16:00 COFFEE BREAK
16:00 – 17:00 LECTURE II – Exercises: Z. Burda
17:00 – 18:00 LECTURE II - Exercises / Solution: J. Bouttier
18:00 – 19:00 LECTURE II – Exercises: T. Jonsson

Wednesday, August 27

9:45 – 11:00 LECTURE III: T. Jonsson
11:00  – 11:30 COFFEE BREAK
11:30  – 12:45 LECTURE III - Exercises / Solution / additionalJ. Bouttier
12:45 – 14:15 LUNCH
14:00 – 15:30 IFT COLLOQUIUM: F. David
15:30 – 16:00 COFFEE BREAK
16:00 – 17:00 POSTER SESSION
17:00 – 18:00 LECTURE III – Exercises: T. Jonsson
18:00 – 19:00 LECTURE III – Exercises: J. Bouttier

Thursday, August 28

9:45 – 11:00 LECTURE I: F. David
11:00  – 11:30 COFFEE BREAK
11:30  – 12:45 LECTURE I: N. Drukker
12:45 – 14:15 LUNCH
14:15 – 15:30 LECTURE IV - Exercises / Solution: J. Bouttier
15:30 – 16:00 COFFEE BREAK
16:00 – 17:00 LECTURE I – Exercises: F. David
17:00 – 18:00 LECTURE I – Exercises: N. Drukker
18:00 – 19:00 LECTURE IV – Exercises: J. Bouttier

Friday, August 29

9:45 – 11:00 LECTURE II: N. Drukker
11:00  – 11:30 COFFEE BREAK
11:30  – 12:45 LECTURE II: F. David
12:45 – 14:15 LUNCH
14:15 – 15:30 LECTURE IV: T. Jonsson
15:30 – 16:00 COFFEE BREAK
16:00 – 17:00 LECTURE II – Exercises: N. Drukker
17:00 – 18:00 LECTURE II – Exercises: F. David
18:00 – 19:00 LECTURE IV – Exercises: T. Jonsson

Saturday and Sunday, August 30 and 31

All day FREE DAY

SECOND WEEK: September 1 and 2

Monday, September 1

9:45 – 11:00 LECTURE III: F. David
11:00  – 11:30 COFFEE BREAK
11:30  – 12:45 LECTURE III: N. Drukker
12:45 – 14:15 LUNCH
14:15 – 15:30 LECTURE III: Z. Burda
15:30 – 16:00 COFFEE BREAK
16:00 – 17:00 LECTURE III – Exercises: F. David
17:00 – 18:00 LECTURE III – Exercises: N. Drukker
18:00 – 19:00 LECTURE III – Exercises: Z. Burda

Tuesday, September 2

9:45 – 11:00 LECTURE IV: Z. Burda
11:00  – 11:30 COFFEE BREAK
11:30  – 12:45 LECTURE IV: F. David
12:45 – 14:15 LUNCH
14:15 – 15:30 LECTURE IV: N. Drukker
15:30 – 16:00 COFFEE BREAK
16:00 – 17:00 LECTURE  IV – Exercises: Z. Burda
17:00 – 18:00 LECTURE IV – Exercises: F. David
18:00 – 19:00 LECTURE IV – Exercises: N. Drukker


Evaluations of school:


Related activities at other Brazilian Institutes: 

Brazilian School of Probability
Mambucaba, Rio de Janeiro – RJ, August 3-9, 2014

Probability in Bahia Meeting
UFBA, Salvador – Bahia, October 6-10, 2014

Further events supported by IMPA 

List of Participants: Updated on August 22

General Information: General Information Sheet -  Useful information specially for those who are not from São Paulo city.

Accommodation: Participants whose accommodation has been arranged and paid by the institute will stay at The Universe FlatEach participant whose accommodation has been arranged by the institute has received the details about the accommodation individually by email.

Registration: ALL participants should register. The registration will be on August 25  from 8:30 to 9:45 at the institute. You can find arrival instruction at http://www.ictp-saifr.org/?page_id=195.

Upon registration, participants who are receiving financial support, please bring a photocopy of your ID or passport.

BOARDING PASS – All participants, whose travel has been provided or will be reimbursed by the institute, should bring the boarding pass upon registration, and collect an envelope to send the return boarding pass to the institute.

Emergency number: 9 8233 8671 (from São Paulo city); +55 11 9 8233 8671 (from abroad), 11 9 8233 8671 (from outside São Paulo).

Ground transportation instructions:

Ground transportation from Guarulhos Airport to The Universe Flat

Ground transportation from The Universe Flat to the institute

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