Research
My current research mainly focuses on the physics and mathematics of twodimensional quantum gravity, as described by random planar maps or Liouville Quantum Gravity. I'm especially interested in the geometry of the random surfaces involved when coupled to matter fields. A central open problem in this study is the determination of the Hausdorff dimension of such surfaces, which is known to be four in the absence of matter. In the presence of matter so far only numerical results, some of which appearing in my papers below, are available, but an analytic understanding, let alone a proof, is lacking.
For a detailed overview of my past research see my research summary [pdf].
Upcoming events (or: where to meet me)
 721 May 2016. Visiting ParisSud math department (LMO), Orsay, France.
 12 May 2016. Seminar at LPT, ParisSud, France.
 610 June 2016. Random trees and maps: probabilistic and combinatorial aspects, CIRM, Marseille, France.
 1620 Jan 2017. Workshop on Random maps and Dimer models, part of the Combinatorics and interactions trimester, IHP, Paris.
 2023 Mar 2017. Quantum gravity in Paris workshop, IHP, Paris.
New
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. One particular application is in the following planar map editor (in progress):
[Planar map editor]
More applications: [Schaeffer's bijection] [Peeling of a uniform infinite triangulation] [Lazy peeling of an infinite planar map] [Lazy peeling of a disk]
Source available on: [github.com]
Squaring of a torus
Any genus1 map, i.e. a graph properly embedded in the torus, gives rise to a twoparameter 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 (spanningtree decorated) genus1 maps.
Papers

[17]Martingales in selfsimilar growthfragmentations and their connections with random planar maps
with J. Bertoin, N. Curien and I. Kortchemski, arXiv1605.00581, [pdf] 
[16]

[15]

[14]The peeling process of infinite Boltzmann planar maps
Electronic journal of combinatorics 23(1) (2016) #P1.28, arXiv:1506.01590, [pdf] 
[13]Multipoint functions of weighted cubic maps
with Jan Ambjorn, Annales de l'Institut Henri Poincaré D 3 (2016) 144, arXiv:1408.3040, [pdf] 
[12]Scaledependent Hausdorff dimensions in 2d gravity
with J. Ambjorn and Y. Watabiki, Phys. Lett. B 736 (2014) 339343, arXiv:1406.6251, [pdf] 
[11]Geodesic distances in quantum Liouville gravity
with Jan Ambjorn, Nucl. Phys. B 889 (2014) 676691, arXiv:1405.3424, [pdf] 
[10]Twodimensional Quantum Geometry
with Jan Ambjorn, Acta Physica Polonica B 44 (2013) 2537, arXiv:1310.8552, [pdf] 
[9]Exploring Torus Universes in Causal Dynamical Triangulations
with Renate Loll, Phys. Rev. D 88 024015 (2013), arXiv:1305.4702 
[8]The toroidal Hausdorff dimension of 2d Euclidean quantum gravity
with Jan Ambjorn, Phys. Lett. B 724 (2013) 328332, arXiv:1305.3674 
[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) 
[6]Semiclassical dynamical triangulations
with Jan Ambjorn, Phys. Lett. B 718 (2012) 200204, arXiv:1209.6031 
[5]

[4]Roaming moduli space using dynamical triangulations
with J. Ambjorn, J. Barkley, Nucl. Phys. B 858 (2012) 267292, arXiv:1110.4649 
[3]Baby Universes Revisited
with J. Ambjorn, J. Barkley and R. Loll, Phys. Lett. B 706 (2011) 8689, arXiv:1110.3998 
[2]Shape Dynamics in 2+1 Dimensions
with Tim Koslowski, Gen. Rel. Grav. 44 (2012) 16151636, arXiv:1107.1287, [pdf] 
[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]
Slides
#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.
Firstpassage 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, ClermontFerrand, 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 nonperturbative 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
Theses
Nonperturbative 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 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 noncritical string theory. We study this connection further by comparing the conformal geometry of dynamical triangulations to analytical results from noncritical 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.
Material:
 Publications [4] and [6]
 Chapter 3 in my PhD thesis
 Slides #2
The Hausdorff dimension of 2d gravity coupled to matter
Material:
 Publications [3], [4], [6] and [8]
Colored triangulations
Material:
 Slides #5
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 socalled 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.
Material:
 Preprint [7]
 Slides #6, #7, and #8
 Random quadrangulations embedded in 3D (requires browser/OS with WEBGL support)
 Movie of random quadrangulations appearing in the cup function.
Miscellaneous
Poster for the CDT & Friends conference
Download high resolution version here.
Nonphysics
Copyright © 2012 Timothy Budd