School on Random Geometry and Random Matrices
Start time: August 25, 2014
Ends on: September 2, 2014
Location: São Paulo, Brazil
Venue: IFTUNESP
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 2day Workshop (September 34). 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 socalled 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, AmbjørnBudd, etc.
3) Distance statistics: the twopoint function, the threepoint function, geodesics…
4) Local limits: UIPT/UIPQ, halfplane 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 nonHermitian 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 nonhermitian 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. 101217, 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. 101217, Elsevier B. V. (2006)
 See the contributions by J. Cardy, W. Werner, I. Kostov & B. Duplantier: Exact Methods in Lowdimensional 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:
http://arxiv.org/abs/arXiv:
http://arxiv.org/abs/arXiv:
Thordur Jonsson (Iceland) – Random tree ensembles and applications
The GaltonWatson 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) 47954811.
 B. Durhuus, T. Jonsson and J. Wheater: The spectral dimension of generic trees. J. Stat. Phys. 128 (2007) 12371260.
 T. Jonsson and S. Ö. Stefánsson: Condensation in nongeneric trees. J. Stat. Phys. 142 (2011) 277313.
 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 / additional: J. 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  

Evaluations of school:
Related activities at other Brazilian Institutes:
Brazilian School of Probability
Mambucaba, Rio de Janeiro – RJ, August 39, 2014
Probability in Bahia Meeting
UFBA, Salvador – Bahia, October 610, 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 Flat. Each 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.ictpsaifr.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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