Random geometry

Programme theme

A new frontier has emerged at the interface between probability, geometry, and analysis, with a central target to produce a coherent theory of the geometry of random structures. The principal question is the following: within a given structure, what is the interplay between randomness and geometry? More precisely, does the geometry appear to be random at every scale (i.e. fractal), or do fluctuations "average out" at sufficiently large scales? Can the global geometry be described by taking a suitable scaling limit that allows for concrete computations?

Spectacular progress has been made over the last ten years in this domain. The goal of the programme is to gather experts from probability, geometry, analysis and other connected areas, in order to study aspects of this question in some paradigmatic situations. Topics of particular relevance include the Gaussian Free Field, random planar maps and Liouville quantum gravity, in connection with conformally invariant scaling limits; spin glass models and branching random walks; percolation and random graphs; and random walks on graphs and groups in the case where the geometry is determined by some algebraic ambient structure.

Click here to download the programme's final scientific report

Preprints

Title	Year	Programme
One-dimensional scaling limits in a planar Laplacian random growth model Authors: Fredrik Viklund, Alan Solar, Amanda Turner Download PDF	2014	RGM	20 February 2019
Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map I: cone times Authors: E Gwynne, C Mao, X Sun Download PDF	2014	RGM	21 October 2016
On the stability of geodesics in the Brownian map Authors: O Angel, B Kolesnik, G Miermont Download PDF	2014	RGM	21 October 2016
The front location in BBM with decay of mass Authors: L Addario-berry Download PDF	2014	RGM	21 October 2016
Joint convergence of random quadrangulations and their cores Authors: L Addario-berry, Y Wen Download PDF	2014	RGM	21 October 2016
A probabilistic approach to block sizes in random maps Authors: L Addario-berry Download PDF	2014	RGM	21 October 2016
Diameter and stationary distributions of random $\it r$-out digraphs Authors: L Addario-berry, B Balle, G Perarnau Download PDF	2014	RGM	21 October 2016
Scaling limits for the critical Fortuin-Kastelyn model on a random planar map II: local estimates and empty reduced word exponent Authors: E Gwynne, X Sun Download PDF	2014	RGM	21 October 2016
Logarithmic coefficients and generalized multifractality of whole-plane SLE Authors: B Duplantier, H Ho, B Le, M Zinsmeister Download PDF	2014	RGM	21 October 2016
A Hamiltonian crycle in the square of a 2-connected graph in linear time Authors: S Alstrup, A Georgakopolous, E Rotenberg, C Thomassen Download PDF	2014	RGM	21 October 2016
Hyperbolicity vs. amenability for planar graphs Authors: B Federici, A Georgakopoulos Download PDF	2014	RGM	21 October 2016

Workshops

RGMW06

RGM follow up

9 July 2018 to 20 July 2018

The Organisers would like to thank the following sponsors for their generous support of the event:

Seminars

Monday 12th January 2015
09:00 to 09:50	Registration	No Room Required
09:50 to 10:00	Welcome from Christie Marr (INI Deputy Director)	No Room Required
10:00 to 11:00	Gaussian Free Field 1	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Random Planar Maps 1	Room 1

Tuesday 13th January 2015
13:30 to 14:30	Schramm-Loewner Evolution 1	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Random Planar Maps 2	Room 1
16:00 to 17:00	Gaussian Free Field 2	Room 1

Wednesday 14th January 2015
09:00 to 10:00	Random Planar Maps 3	Room 1
10:00 to 11:00	Schramm-Loewner Evolution 2	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Office hours	No Room Required

Thursday 15th January 2015
13:30 to 14:30	Schramm-Loewner Evolution 3	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Gaussian Free Field 3	Room 1
16:00 to 17:00	Random Planar Maps 4	Room 1
19:30 to 22:00	Conference Dinner at Emmanuel College	No Room Required

Friday 16th January 2015
09:00 to 10:00	Discrete Lattice Models 1	Room 1
10:00 to 11:00	Discrete Lattice Models 2	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Office hours	No Room Required

Monday 19th January 2015
10:00 to 11:00	Gaussian Multiplicative Chaos 1	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Gaussian Free Field 4	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
13:30 to 14:30	Gaussian Multiplicative Chaos 2	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Discrete Lattice Models 3	Room 1

Tuesday 20th January 2015
09:00 to 10:00	Schramm-Loewner Evolution 4	Room 1
10:00 to 11:00	Gaussian Multiplicative Chaos 3	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Office hours	No Room Required
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
13:30 to 14:30	Gaussian Multiplicative Chaos 4	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Gaussian Free Field 5	Room 1

Wednesday 21st January 2015
09:00 to 10:00	Discrete Lattice Models 4	Room 1
10:00 to 11:00	Schramm-Loewner Evolution 5	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Random Planar Maps 5	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
13:30 to 14:30	Gaussian Multiplicative Chaos 5	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Gaussian Multiplicative Chaos 6	Room 1

Thursday 22nd January 2015
09:00 to 10:00	Schramm-Loewner Evolution 6	Room 1
10:00 to 11:00	Random Planar Maps 6	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Gaussian Multiplicative Chaos 6	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
13:30 to 14:30	Office hours	No Room Required
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Discrete Lattice Models 5	Room 1

Friday 23rd January 2015
09:00 to 10:00	Gaussian Free Field 7	Room 1
10:00 to 11:00	Gaussian Multiplicative Chaos 7	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Discrete Lattice Models 6	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
13:30 to 14:30	Schramm-Loewner Evolution 7	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Random Planar Maps 7	Room 1

Monday 26th January 2015
09:00 to 09:50	Registration	No Room Required
09:50 to 10:00	Welcome from John Toland (INI Director)	No Room Required
10:00 to 11:00	Renormalization via merging trees	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	The extremal process in nested conformal loops	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
13:30 to 14:30	Ising Model, Conformal Field Theory, etc	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Aperiodic hierarchical conformal tilings: random at the ends?	Room 1
17:00 to 18:00	Welcome Wine Reception	No Room Required

Tuesday 27th January 2015
09:00 to 10:00	Parafermionic observables and order of the phase transition in planar random-cluster models	Room 1
10:00 to 11:00	Liouville Quantum Gravity on the Riemann sphere	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	tba	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
13:30 to 14:30	A conformally invariant metric on CLE(4)	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Conformal invariance of boundary touching loops of FK Ising model	Room 1

Wednesday 28th January 2015
09:00 to 10:00	SLE correlations and singular vectors	Room 1
10:00 to 11:00	Almost sure multifractal spectrum of SLE	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	From the critical Ising model to spanning trees	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
13:30 to 14:30	GFF with SLE and KPZ	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 16:00	Generalized Multifractality of Whole-Plane SLE	Room 1
19:30 to 22:00	Conference Dinner at Cambridge Union Society hosted by Cambridge Dining Co.	No Room Required

Thursday 29th January 2015
09:00 to 10:00	Liouville quantum gravity as a mating of trees	Room 1
10:00 to 11:00	Geodesics in Brownian surfaces	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Drawing maps	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
13:30 to 14:30	Some scaling limit results for critical Fortuin-Kastelyn random planar map model	Room 1

Friday 30th January 2015
09:00 to 10:00	Scaling limits of critical Ising correlation functions in planar domains	Room 1
10:00 to 11:00	Renormalization Approach to the 2-Dimensional Uniform Spanning Tree	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	The Z-invariant massive Laplacian on isoradial graphs	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
13:30 to 14:30	On near-critical SLE(6) and on the tail in Cardy's formula	Room 1

Monday 2nd February 2015
16:00 to 17:00	Rothschild Distinguished Visiting Fellow Lecture: Random maps and random 2-dimensional geometries	Room 1
17:00 to 18:00	Wine Reception	No Room Required

Thursday 5th February 2015
16:00 to 17:00	The dynamic phi^4 model in the plane	Room 2

Thursday 26th February 2015
10:00 to 12:00	RGM Discussion Group	Room 2

Friday 27th February 2015
14:00 to 15:00	informal discussion	Room 2

Wednesday 11th March 2015
12:30 to 13:30	On random Hamilton-Jacobi equation and "other" KPZ	Room 1

Monday 16th March 2015
09:00 to 09:50	Registration	No Room Required
09:50 to 10:00	Welcome from Christie Marr (INI Deputy Director)	Room 1
10:00 to 11:00	Compensated Fragmentations	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Random trees constructed by aggregation	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	Google maps and improper Poisson line processes	Room 1
15:00 to 15:30	Afternoon Tea	No Room Required
15:30 to 16:30	A line-breaking construction of the stable trees	Room 1
17:00 to 18:00	Welcome Wine Reception	No Room Required

Tuesday 17th March 2015
10:00 to 11:00	The Compulsive Gambler process	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Delocalization of two-dimensional random surfaces with hard-core constraints	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	Small-particle limits in a regularized Laplacian random growth model	Room 1
15:00 to 15:30	Afternoon Tea	No Room Required
15:30 to 16:30	Local graph coloring	Room 1

Wednesday 18th March 2015
10:00 to 11:00	Branching Brownian motion, the Brownian net and selection in spatially structured populations	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Rigorous results for a population model with selection	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	Genealogies with recombination in spatial population genetics	Room 1
15:00 to 15:30	Afternoon Tea	No Room Required
15:30 to 16:30	Branching processes with competition by pruning of Levy trees	Room 1
19:30 to 22:00	Conference Dinner at Christ's College	No Room Required

Thursday 19th March 2015
10:00 to 11:00	A multi-scale refinement of the second moment method Co-authors: L.P	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Log-correlated Gaussian fields: study of the Gibbs measure	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	On the complex cascade and the complex branching Brownian motion	Room 1
15:00 to 15:30	Afternoon Tea	No Room Required
15:30 to 16:30	Maxima of log-correlated Gaussian fields and of the Riemann Zeta function on the critical line	Room 1

Friday 20th March 2015
10:00 to 11:00	Scale-free percolation	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Rate of convergence of the mean of sub-additive ergodic processes	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	Uniformity of the late points of random walk on $\mathbb{Z}_d^n$ for $d \geq 3$	Room 1
15:00 to 15:30	Afternoon Tea	No Room Required
15:30 to 16:30	Maxima of logarithmically correlated fields	Room 1

Wednesday 25th March 2015
10:00 to 12:00	RGM Discussion Group	Room 2

Wednesday 1st April 2015
10:00 to 12:00	RGM Discussion Group	Room 2

Wednesday 8th April 2015
10:00 to 12:00	RGM Discussion Group	Room 2

Friday 10th April 2015
14:30 to 16:30	Discussion booked by Christina Goldschmidt	Discussion Room

Wednesday 15th April 2015
10:00 to 12:00	RGM Discussion Group	Room 2

Monday 20th April 2015
09:00 to 09:50	Registration	No Room Required
09:50 to 10:00	Welcome from Christie Marr (INI Deputy Director)	Room 1
10:00 to 11:00	Compact Brownian surfaces	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Blossoming trees and the scaling limit of maps	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	Scaling constants and the lazy peeling of infinite Boltzmann planar maps	Room 1
15:00 to 15:30	Afternoon Tea	No Room Required
15:30 to 16:30	Critical exponents in FK-weighted planar maps	Room 1
16:30 to 17:30	Welcome Wine Reception	No Room Required

Tuesday 21st April 2015
10:00 to 11:00	Scaling limits of the uniform spanning tree	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	A hidden quantum group for pure partition functions of multiple SLEs	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 16:00	Discussion	Room 2
14:00 to 15:00	Self-avoiding Walk and Connective Constant	Room 1
15:00 to 15:30	Afternoon Tea	No Room Required
15:30 to 16:30	Looptrees	Room 1

Wednesday 22nd April 2015
09:00 to 10:00	On the geometry of discrete and continuous random planar maps	Room 1
10:00 to 11:00	On classes of planar maps with $\alpha$-orientations having geometric interpretations	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Nesting statistics in the O(n) loop model on random lattices	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	Differential equations for colored maps	Room 1
15:00 to 15:30	Afternoon Tea	No Room Required
15:30 to 16:30	Recurrence of planar graph limits	Room 1
19:30 to 22:00	Conference Dinner at Trinity College	No Room Required

Thursday 23rd April 2015
09:00 to 10:00	The uniform spanning forest of planar graphs	Room 1
10:00 to 11:00	Characterstic polynomials of random matrices and logarithmically correlated processes	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Pinning and disorder relevance for the lattice Gaussian free field	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	Planar lattices do not recover from forest fires	Room 1
15:00 to 15:30	Afternoon Tea	No Room Required
15:30 to 16:30	The exact $k$-SAT threshold for large $k$	Room 1

Friday 24th April 2015
10:00 to 11:00	Squarings of rectangles	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Parabolic and Hyperbolic Unimodular maps	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	Liouville Quantum gravity on Riemann surfaces	Room 1
15:00 to 15:30	Afternoon Tea	No Room Required
15:30 to 16:30	Scaling limits of random planar maps and growth-fragmentations	Room 1

Wednesday 6th May 2015
10:00 to 12:00	RGM Discussion Group	Room 2

Wednesday 13th May 2015
10:00 to 12:00	RGM Discussion Group	Room 2

Wednesday 20th May 2015
10:00 to 12:00	RGM Discussion Group	Room 2

Wednesday 27th May 2015
10:00 to 12:00	RGM Discussion Group	Room 2

Wednesday 3rd June 2015
10:00 to 12:00	RGM Discussion Group	Room 2

Wednesday 10th June 2015
10:00 to 12:00	RGM Discussion Group	Room 2

Monday 15th June 2015
09:00 to 09:50	Registration	No Room Required
09:50 to 09:55	Welcome from Christie Marr (INI Deputy Director)	No Room Required
09:55 to 10:00	Welcome from Organisers	No Room Required
10:00 to 11:00	Some news from the loop-soup front	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Conformal representations of Random Maps and Surfaces	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	Liouville quantum gravity and the Brownian map	Room 1
15:00 to 15:30	Afternoon Tea	No Room Required
15:30 to 16:30	Essential spanning forests on periodic planar graphs	Room 1
16:30 to 17:30	Welcome Wine Reception	No Room Required

Tuesday 16th June 2015
09:00 to 10:00	Competitive erosion is conformally invariant	Room 1
10:00 to 10:10	Break	No Room Required
10:10 to 11:10	On Disconnection, random walks, random interlacements, and the Gaussian free field	Room 1
11:10 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	tba	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	Convergence of discrete holomorphic functions on non-uniform lattices	Room 1
15:00 to 15:30	Afternoon Tea	No Room Required
15:30 to 16:30	Conformal restriction: the chordal and the radial	Room 1

Wednesday 17th June 2015
09:00 to 10:00	Scaling window of Bernoulli percolation on Z^d	Room 1
10:00 to 10:10	Break	No Room Required
10:10 to 11:10	A (slightly) new look at the backbone	Room 1
11:10 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	A random walk proof of Kirchhoff's matrix tree theorem	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	Return of the Multiplicative Coalescent	Room 1
15:00 to 15:30	Afternoon Tea	No Room Required
15:30 to 16:30	Where Planar Simple Random Walk Loses its Rotational Symmetry	Room 1
19:30 to 22:00	Conference Dinner at Gonville and Caius College	No Room Required

Thursday 18th June 2015
10:00 to 11:00	SLE Quantum Multifractality	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Radial SLE martingale-observables	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	Scaling limit of the probability that loop-erased random walk uses a given edge	Room 1
15:00 to 15:20	Afternoon Tea	No Room Required
15:20 to 16:20	Welding of the Backward SLE and Tip of the Forward SLE	Room 1
16:20 to 16:30	Break	No Room Required
16:30 to 17:30	Boundary Measures and Natural Time Parameterization for SLE	Room 1

Friday 19th June 2015
10:00 to 11:00	Twin peaks	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:30	Loewner curvature	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
14:00 to 15:00	From Internal DLA to self-interacting walks	Room 1

Tuesday 23rd June 2015
15:00 to 17:00	RGM Discussion Group	Room 2

Wednesday 24th June 2015
10:00 to 12:00	RGM Discussion Group	Room 2

Wednesday 1st July 2015
10:00 to 12:00	RGM Discussion Group	Room 2

Organisers

Itai Benjamini University of California, Irvine and Weizmann Institute of Science
Nathanael Berestycki University of Cambridge
Jean-Francois Le Gall Université Paris Saclay
Scott Sheffield Massachusetts Institute of Technology

Programme theme

Preprints

One-dimensional scaling limits in a planar Laplacian random growth model

Scaling limits for the critical Fortuin-Kasteleyn model on a random planar map I: cone times

On the stability of geodesics in the Brownian map

The front location in BBM with decay of mass

Joint convergence of random quadrangulations and their cores

A probabilistic approach to block sizes in random maps

Diameter and stationary distributions of random $\it r$-out digraphs

Scaling limits for the critical Fortuin-Kastelyn model on a random planar map II: local estimates and empty reduced word exponent

Logarithmic coefficients and generalized multifractality of whole-plane SLE

A Hamiltonian crycle in the square of a 2-connected graph in linear time

Hyperbolicity vs. amenability for planar graphs

Workshops

RGM follow up

Seminars

Organisers

Supported By

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Logarithmic coefficients and generalized multifractality of whole-plane SLE