Timothy Budd

Timothy Budd

Assistant professor
Theoretical Particle Physics and Cosmology
Niels Bohr Institute
Copenhagen University
Fc-10 (second floor above the canteen in the F building)
My CV [pdf]


My current research mainly focuses on the physics and mathematics of two-dimensional 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)


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.
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]
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.


  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, 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 [INSPIRE] [arXiv] [Google Scholar]


Geometry of random planar maps with high degrees
28 Apr 2016. Seminar at UMPA, ENS de Lyon, France.
The peeling process on random planar maps with loops
3 Dec 2015. Séminaire Philippe Flajolet, l'Institut Henri Poincaré, Paris, France.
Peeling of infinite Boltzmann planar maps
20th Itzykson conference, IPTh, Saclay, France, 12 June 2015
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
First-passage percolation on random planar maps
Probability on Trees and Planar Graphs, Banff International Research Station, Banff, Canada, 15 Sept. 2014
Relating discrete and continuum 2d quantum gravity
Quantum gravity seminar, Radboud University Nijmegen, The Netherlands, 14 Apr. 2014
Fractal dimensions of 2d quantum gravity
Approaches to Quantum Gravity, Meeting of GDR, Université Blaise Pascal, Clermont-Ferrand, France, 6 Jan. 2014
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
Generalized CDT as a scaling limit of planar maps
Invited talk at Quantum Gravity in Paris, Orsay, France, 20 Mar. 2013
CDT and trees
Invited talk at the CDT and Friends conference, Nijmegen, The Netherlands, 14 Dec. 2012
Adding colors to dynamical triangulations in 3d
Seminar, Niels Bohr Institute, Copenhagen, 20 Sep. 2012
Effective dynamics in non-perturbative quantum gravity
QUIST/thesis seminar, ITP, Utrecht, 15 Mar. 2012
Effective dynamics of CDT in 2+1 dimensions
Quantum gravity seminar, Perimeter Institute, Waterloo, Canada, 14 Dec. 2011
Probing moduli space using dynamical triangulations
GRAFITI seminar, ITP, Utrecht, 7 Nov. 2011
The effective kinetic term in CDT
LOOPS `11, Madrid, Spain, 24 May 2011


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


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.


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


Some of my recent photos
View them on