Timothy Budd

Timothy Budd

Postdoctoral researcher
Institut de Physique Théorique
CEA, Université Paris-Saclay
Email
timothy.budd@cea.fr
Office
Bâtiment 774, P 025
ORCID
0000-0002-1130-0178
My CV [pdf]

Research

My research interests lie at the interface of theoretical physics, probability theory, combinatorics and algorithms. Research topcis include: (two-dimensional) quantum gravity, general relativity, statistical physics, random planar maps, random walks, stochastic processes, and more.

For an overview of earlier research see my research summary [pdf].

Upcoming events

Teaching: Analytic Combinatorics and applications

In February and March of 2017 I'm teaching a Master's course on Analytic combinatorics as part of the M2 program on probability at Le Département de Mathématiques d'Orsay, Université de Paris-Sud.

Lectures will take place on Thursday mornings 9:00-12:00 in room 117-119 of Bâtiment 425 (math department) of the Orsay campus. First lecture will be on February 2nd (so no lecture yet on January 26th!). The lectures will mainly follow the excellent book of Flajolet and Sedgewick (F&S), "Analytic Combinatorics", Cambridge University Press, which is freely downloadable. The schedule for the course, with some keywords, is as follows:

February 2nd
Lecture 1: Formal power series; Combinatorial classes; Ordinary generating functions; Iterative and recursive constructions; Plane trees. (F&S, Chapter I)
February 9th
Lecture 2: Labelled combinatorial classes; Exponential generating functions (F&S, Chapter II); Review of various concepts of complex analysis.
February 16th
No Lecture (winter holiday).
February 23rd
Lecture 3: Complex analysis; singularities of generating functions; transfer theorems (F&S, Chapter IV + VI).
March 2nd
Lecture 4: Multivariate generating functions; limit laws; planar maps and quadratic method.
March 9th
Lecture 5: Saddle-point asymptotics; ....
March 16th
Lecture 6: (Student presentations)
March 23rd
Lecture 7: (Student presentations)

Interactive stuff

Infinite planar maps with high degrees
A portion of an infinite Boltzmann planar map with vertices of high degree (blue balls). Shown are also a number of geodesics which tend to merge in high degree vertices. The geometry of these objects is the subject of the recent paper [15] with Nicolas Curien.
Planarmap.js
A Javascript library in progress aimed at visualizing and interacting with planar maps in the browser. Any suggestions and improvements are more than welcome. So far the most useful application seems to be this editor (in progress): [Planarmap.js editor]
Squaring of a torus
Any genus-1 map, i.e. a graph properly embedded in the torus, gives rise to a two-parameter family of tilings of the torus by squares of various sizes. The linked webpage allows users to explore such square tilings associated to randomly generated (spanning-tree decorated) genus-1 maps.

Papers

  1. [17]
    Martingales in self-similar growth-fragmentations and their connections with random planar maps
    with J. Bertoin, N. Curien and I. Kortchemski, arXiv1605.00581, [pdf]
  2. [16]
    Generalized multicritical one-matrix models
    with J. Ambjorn and Y. Makeenko, Nucl. Phys. B 913 (2016) 357-380, arXiv:1604.04522
  3. [15]
    Geometry of infinite planar maps with high degrees
    with Nicolas Curien, arXiv:1602.01328, [pdf]
  4. [14]
    The peeling process of infinite Boltzmann planar maps
    Electronic journal of combinatorics 23(1) (2016) #P1.28, arXiv:1506.01590, [pdf]
  5. [13]
    Multi-point functions of weighted cubic maps
    with Jan Ambjorn, Annales de l'Institut Henri Poincaré D 3 (2016) 1-44, arXiv:1408.3040, [pdf]
  6. [12]
    Scale-dependent Hausdorff dimensions in 2d gravity
    with J. Ambjorn and Y. Watabiki, Phys. Lett. B 736 (2014) 339-343, arXiv:1406.6251, [pdf]
  7. [11]
    Geodesic distances in quantum Liouville gravity
    with Jan Ambjorn, Nucl. Phys. B 889 (2014) 676-691, arXiv:1405.3424, [pdf]
  8. [10]
    Two-dimensional Quantum Geometry
    with Jan Ambjorn, Acta Physica Polonica B 44 (2013) 2537, arXiv:1310.8552, [pdf]
  9. [9]
    Exploring Torus Universes in Causal Dynamical Triangulations
    with Renate Loll, Phys. Rev. D 88 024015 (2013), arXiv:1305.4702
  10. [8]
    The toroidal Hausdorff dimension of 2d Euclidean quantum gravity
    with Jan Ambjorn, Phys. Lett. B 724 (2013) 328-332, arXiv:1305.3674
  11. [7]
    Trees and spatial topology change in CDT
    with Jan Ambjorn, J. Phys. A: Math. Theor. 46 (2013) 315201, arXiv:1302.1763 (featured on journal front cover)
  12. [6]
    Semi-classical dynamical triangulations
    with Jan Ambjorn, Phys. Lett. B 718 (2012) 200-204, arXiv:1209.6031
  13. [5]
    The effective kinetic term in CDT
    J. Phys.: Conf. Ser. 360 (2012) 012038, arXiv:1110.5158
  14. [4]
    Roaming moduli space using dynamical triangulations
    with J. Ambjorn, J. Barkley, Nucl. Phys. B 858 (2012) 267-292, arXiv:1110.4649
  15. [3]
    Baby Universes Revisited
    with J. Ambjorn, J. Barkley and R. Loll, Phys. Lett. B 706 (2011) 86-89, arXiv:1110.3998
  16. [2]
    Shape Dynamics in 2+1 Dimensions
    with Tim Koslowski, Gen. Rel. Grav. 44 (2012) 1615-1636, arXiv:1107.1287, [pdf]
  17. [1]
    In Search of Fundamental Discreteness in 2+1 Dimensional Quantum Gravity
    with Renate Loll, Class. Quant. Grav. 26 (2009) 185011, arXiv:0906.3547, [pdf]
Overview on [arXiv] [Google Scholar] [INSPIRE]

Talks and Slides

▼ Show/hide full list of talks ▼
15 May 2017
Peeling of random planar maps, Mini-course at Mini-school on Random Maps and the Gaussian Free Field, École normale supérieure de Lyon.
28 Apr 2017
Combinatorics seminar at the Mathematics Institute, University of Warwick, UK.
24 Apr 2017
Computations and Proofs seminar at SpecFun, INRIA, École Polytechnique, Palaiseau, France.
20 Mar 2017
Invited conference talk at Quantum gravity in Paris, IHP, Paris.
10 Mar 2017
Seminar at the Laboratoire de Mathématiques et Physique Théorique, Université de Tours, France.
25 Jan 2017
Winding of walks on the square lattice, Invited conference talk at Journées de combinatoire de Bordeaux, LaBRI, Bordeaux. [slides]
19 Jan 2017
On a connection between planar map combinatorics and lattice walks, Invited conference talk at Workshop on Random maps and Dimer models, part of the Combinatorics and interactions trimester, IHP, Paris. [slides]
7 Jun 2016
Geometry of random planar maps with high degrees, Invited conference talk at Random trees and maps: probabilistic and combinatorial aspects, CIRM, Marseille.
12 May 2016
Geometry of random planar maps with high degrees, Seminar at LPT, Paris-Sud.
28 Apr 2016
Geometry of random planar maps with high degrees, Seminar at UMPA, ENS de Lyon. [slides]
20 Apr 2016
QMATH Quantum Lunch seminar, Math department, University of Copenhagen.
17 Mar 2016
Random planar maps, outreach talk for students participating in a study trip organized by A-Eskwadraat, University of Copenhagen.
7 Mar 2016
On the universality of 2d quantum gravity, Journal club, Niels Bohr Institute, University of Copenhagen.
3 Dec 2015
The peeling process on random planar maps with loops, Invited conference talk at Séminaire Philippe Flajolet, l'Institut Henri Poincaré, Paris. [slides]
26 Nov 2015
From random walks to loops on random planar maps, Seminar at LMO, Orsay, France.
24 Jun 2015
The O(n) model on random surfaces, Journal club, Niels Bohr Institute, University of Copenhagen.
12 Jun 2015
Peeling of infinite Boltzmann planar maps, Invited conference talk at 20th Itzykson conference, IPTh, Saclay, France.
20 Apr 2015
Scaling constants and the lazy peeling of infinite Boltzmann planar maps, Invited conference talk at Random Planar Structures and Statistical Mechanics, Isaac Newton Institute, Cambridge. [slides], [video]
15 Sep 2014
First-passage percolation on random planar maps, Invited conference talk at Probability on Trees and Planar Graphs, Banff International Research Station, Banff, Canada. [slides], [video]
14 Apr 2014
Relating discrete and continuum 2d quantum gravity, Quantum gravity seminar, Radboud University Nijmegen, The Netherlands. [slides]
6 Jan 2014
Fractal dimensions of 2d quantum gravity, Invited conference talk at Approaches to Quantum Gravity, Meeting of GDR, Université Blaise Pascal, Clermont-Ferrand, France. [slides]
20 Jun 2013
From planar maps to spatial topology change in 2d gravity, Invited conference talk at Journées Cartes, l'Institut de Physique Théorique, CEA Saclay, France. [slides]
20 Mar 2013
Generalized CDT as a scaling limit of planar maps, Invited conference talk at Quantum Gravity in Paris, Orsay, France. [slides]
14 Dec 2012
CDT and trees, Invited conference talk at the CDT and Friends, Nijmegen, The Netherlands. [slides]
20 Sep 2012
Adding colors to dynamical triangulations in 3d, Seminar, Niels Bohr Institute, Copenhagen. [slides]
15 Mar 2012
Effective dynamics in non-perturbative quantum gravity, PhD thesis seminar, ITP, Utrecht. [slides]
14 Dec 2011
Effective dynamics of CDT in 2+1 dimensions, Seminar, Perimeter Institute, Waterloo, Canada. [slides]
7 Nov 2011
Probing moduli space using dynamical triangulations, Seminar, ITP, Utrecht. [slides]
24 May 2011
The effective kinetic term in CDT, conference talk at LOOPS `11, Madrid, Spain. [slides]
30 Nov 2010
Having fun with cute little universes, PhD lunch seminar, ITP, Utrecht.
2 Feb 2010
Conformal trouble in quantum gravity, talk at PhD School on Theoretical High Energy Physics, Driebergen, The Netherlands.
9 Jan 2009
Geometric observables in 2+1 dimensional quantum gravity, ENRAGE Young quantum gravity meeting, Imperial college, London.

 

#17.
Winding of walks on the square lattice
25 Jan 2017. Journées de combinatoire de Bordeaux, LaBRI, Bordeaux.
#16.
On a connection between planar map combinatorics and lattice walks
#15.
Geometry of random planar maps with high degrees
28 Apr 2016. Seminar at UMPA, ENS de Lyon, France.
#14.
The peeling process on random planar maps with loops
3 Dec 2015. Séminaire Philippe Flajolet, l'Institut Henri Poincaré, Paris, France.
#13.
Peeling of infinite Boltzmann planar maps
20th Itzykson conference, IPTh, Saclay, France, 12 June 2015
#12.
Scaling constants and the lazy peeling of infinite Boltzmann planar maps
Random Planar Structures and Statistical Mechanics, Isaac Newton Institute, Cambridge, UK, 20 April 2015
#11.
First-passage percolation on random planar maps
Probability on Trees and Planar Graphs, Banff International Research Station, Banff, Canada, 15 Sept. 2014
#10.
Relating discrete and continuum 2d quantum gravity
Quantum gravity seminar, Radboud University Nijmegen, The Netherlands, 14 Apr. 2014
#9.
Fractal dimensions of 2d quantum gravity
Approaches to Quantum Gravity, Meeting of GDR, Université Blaise Pascal, Clermont-Ferrand, France, 6 Jan. 2014
#8.
From planar maps to spatial topology change in 2d gravity
Invited talk at Journées Cartes, l'Institut de Physique Théorique, CEA Saclay, France, 20 Jun. 2013
#7.
Generalized CDT as a scaling limit of planar maps
Invited talk at Quantum Gravity in Paris, Orsay, France, 20 Mar. 2013
#6.
CDT and trees
Invited talk at the CDT and Friends conference, Nijmegen, The Netherlands, 14 Dec. 2012
#5.
Adding colors to dynamical triangulations in 3d
Seminar, Niels Bohr Institute, Copenhagen, 20 Sep. 2012
#4.
Effective dynamics in non-perturbative quantum gravity
QUIST/thesis seminar, ITP, Utrecht, 15 Mar. 2012
#3.
Effective dynamics of CDT in 2+1 dimensions
Quantum gravity seminar, Perimeter Institute, Waterloo, Canada, 14 Dec. 2011
#2.
Probing moduli space using dynamical triangulations
GRAFITI seminar, ITP, Utrecht, 7 Nov. 2011
#1.
The effective kinetic term in CDT
LOOPS `11, Madrid, Spain, 24 May 2011

Theses

Non-perturbative quantum gravity: a conformal perspective
PhD thesis defended on 20 March 2012, Utrecht University, The Netherlands. Supervisor: Prof. R. Loll
Geometric observables in 2+1 dimensional quantum gravity
Master thesis defended on 31 August 2007, Utrecht University, The Netherlands. Supervisor: Prof. R. Loll

Videos

Overview of all videos on [Youtube].
Random square tiling of the plane
Circle packed peeling of a random triangulation
2D gravity from circle patterns
Geodesic distance in quantum Liouville gravity
Zooming in on 2D Quantum Gravity
Semi-classical dynamical triangulation
Dynamical triangulation of the 2-torus
Random quadrangulations
This movie displays some random quadrangulations appearing in the disk amplitudes of generalized CDT.

3d models of 2d gravity

See all models on [Sketchfab]
Brownian map
A random quadrangulation of the sphere with 10k quadrangles.
Brownian map
A random quadrangulation of the sphere with 20k quadrangles.
Brownian map
A random quadrangulation of the sphere with 32k quadrangles.
CDT disk
A random causal triangulation of a disk.
Generalized CDT
A random quadrangulation of the sphere with a limited number of local maxima of the distance function to a marked vertex.

Older projects

CDT in 2+1 dimensions: going beyond spatial volume measurements
Most of what is known about the dynamics of Causal Dynamical Triangulations (CDT) has been deduced from measurements of the spatial volume as function of time. However, the amount of information in these measurements is fundamentally limited. In particular, to address the question of what continuum model CDT reduces to in its (semi)classical limit, it is necessary to study observables which capture different aspects of the spatial geometry. The quest for such observables has lead us to study universes with spatial topology of the 2d torus, where certain observables, known as moduli, can be given a rigorous definition and can be studied numerically.
Conformal geometry in 2d Dynamical triangulations
There is substantial evidence that dynamical triangulations in two dimensions provides a discretization of non-critical string theory. We study this connection further by comparing the conformal geometry of dynamical triangulations to analytical results from non-critical string theory. In particular, we find good numerical agreement between the distributions of moduli parameters of randomly triangulated tori coupled to various matter fields and corresponding quantities in 2d Liouville gravity.
The Hausdorff dimension of 2d gravity coupled to matter
Colored triangulations
Spatial topology change and tree bijections in 2d
In causal dynamical triangulations the spatial topology of the universe is not allowed to change in time. However, at least in two dimensions, it is possible to incorporate sporadic topology changes while maintaining a sensible continuum limit, leading to so-called generalized CDT. We demonstrate how one can study this model by taking the continuum limit of random quadrangulations. The analysis relies heavily on bijections between quadrangulations and labeled trees introduced Cori, Vauquelin and Schaeffer.

Miscellaneous

Poster for the CDT & Friends conference
Download high resolution version here.

Non-physics

Some of my recent photos
View them on