Random Geometry by Cambridge University

What's Random Geometry about?

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.

What's Random Geometry about?

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Random Geometry episodes:

The master field in the plane

Boundary Measures and Natural Time Parameterization for SLE

Twin peaks

From Internal DLA to self-interacting walks

Radial SLE martingale-observables

SLE Quantum Multifractality

Loewner curvature

Scaling limit of the probability that loop-erased random walk uses a given edge

Welding of the Backward SLE and Tip of the Forward SLE

Return of the Multiplicative Coalescent

Where Planar Simple Random Walk Loses its Rotational Symmetry

A (slightly) new look at the backbone

Scaling window of Bernoulli percolation on Z^d

A random walk proof of Kirchhoff's matrix tree theorem

Convergence of discrete holomorphic functions on non-uniform lattices

Double dimers

On Disconnection, random walks, random interlacements, and the Gaussian free field

Some news from the loop-soup front

Competitive erosion is conformally invariant

Liouville quantum gravity and the Brownian map

Conformal restriction: the chordal and the radial

Conformal representations of Random Maps and Surfaces

Essential spanning forests on periodic planar graphs

Scaling limits of random planar maps and growth-fragmentations

Parabolic and Hyperbolic Unimodular maps

Liouville Quantum gravity on Riemann surfaces

Squarings of rectangles

The exact k-SAT threshold for large k

Pinning and disorder relevance for the lattice Gaussian free field

The uniform spanning forest of planar graphs

Characterstic polynomials of random matrices and logarithmically correlated processes

Planar lattices do not recover from forest fires

Differential equations for colored maps

On the geometry of discrete and continuous random planar maps

On classes of planar maps with α-orientations having geometric interpretations

Recurrence of planar graph limits

Looptrees

A hidden quantum group for pure partition functions of multiple SLEs

Scaling limits of the uniform spanning tree

Compact Brownian surfaces

Scaling constants and the lazy peeling of infinite Boltzmann planar maps

Critical exponents in FK-weighted planar maps

Blossoming trees and the scaling limit of maps

Scale-free percolation

Rate of convergence of the mean of sub-additive ergodic processes

Uniformity of the late points of random walk on Znd for d≥3

Maxima of log-correlated Gaussian fields and of the Riemann Zeta function on the critical line

On the complex cascade and the complex branching Brownian motion

A multi-scale refinement of the second moment method Co-authors: L.P

Log-correlated Gaussian fields: study of the Gibbs measure