Oded SchrammOded was in the Theory Group of Microsoft Research. Publication list (based on Bibserver). Short CV. |
To our profound sadness and shock, our colleague and friend, Oded Schramm, died in a tragic hiking accident on September 1, 2008.
Oded was a towering figure, an extraordinary mathematician, widely considered to be the most influential probabilist in the world. His revolutionary work transformed our understanding of critical processes in two dimensions through his introduction of the Stochastic Loewner evolution, tying probability theory to complex analysis in a completely novel way. He also made fundamental contributions to circle packings, random spanning trees, percolation, noise sensitivity of Boolean functions, random permutations and metric geometry.
Oded worked at Microsoft Research for the last ten years. He received the Erdős Prize in Mathematics in 1996, the Salem Prize in 2001, the Clay Research Award in 2002, the Poincaré Prize in 2003, the Loève Prize in 2003, the Pólya Prize in 2006 and the Ostrowski Prize in 2007. He was elected as a member of the Royal Swedish Academy of Sciences in 2008. Oded gave many key lectures, including plenary addresses in the 2004 European Congress of Mathematics and the 2006 International Congress of Mathematicians, as well as the 2005 Coxeter Lecture Series at the Fields Institute and the 2006 Abel lecture. On the theory group webpage, Oded listed his interests:
Oded was a remarkable individual: always calm, humble, generous with his insights and ideas, the best collaborator one could hope for and the person who could always be relied upon. Our heart is with Oded’s family. He will be sorely missed by all who knew him. Due to the outpouring of emotion at the news of Oded's untimely death, we created this memorial web page and the Oded Schramm Memorial Blog to allow his friends to share their photographs and memories of him. There will be a workshop held in Oded's honor on August 30-31, 2009, at Microsoft Research. |
Percolation, Brownian Motion and SLE (Yale Whittemore Lectures, 2001)
The Scaling Limit of Loop-Erased Random Walk (2002)
Scaling Limits of Random Processes and the Outer
Boundary of Planar Brownian Motion (2000)
Abstract: Consider a random walk on the square grid in the plane: a particle is placed at the origin, and at each step moves to a random vertex adjacent to the current position, with all choices having equal probability. If one performs this process on finer and finer grids, and rescales time appropriately, the process converges to a random path, which is called Brownian motion. Perhaps surprisingly, Brownian motion is more symmetric than the random walk: it has rotational symmetry. In fact, it is conformally invariant, which is a more general kind of symmetry. A more complicated process is critical percolation, where each edge of the square grid is deleted with probability 1/2, independently, and the connectivity properties of the resulting graph are studied. It is an outstanding challenge to understand what happens to critical percolation and similar processes when the mesh of the grid tends to zero. Many of these processes are believed to display conformal invariance in the limit, but this is mostly unproven. Under the assumption of conformal invariance we give a complete description of the scaling limit of critical percolation and several other models. The description is based on a process that we call Stochastic Loewner Evolution (SLE). The SLE process describes a randomly growing set by specifying the conformal map to the complement of the set. The conformal map is obtained by solving a random differential equation. There's one free parameter κ> 0 in the description of SLE. In joint work with Greg Lawler and Wendelin Werner, we prove that many properties of planar Brownian Motion are the same as those of SLE with κ=6. This is then used to answer several problems regarding planar Brownian Motion. In particular, we prove Mandelbrot's conjecture stating that the Hausdorff dimension of the outer boundary of planar Brownian Motion is 4/3. The talk will assume no prior knowledge. The plan is to describe some of the random processes, explain and motivate the construction of SLE, and explain the relation with planar Brownian Motion. |
Two demonstrations of loop-erased random walk: coarse grid and fine grid (both are PostScript files that should be paged through).
A demonstration of SLE for Windows (written by David B. Wilson)
A demonstration of Wilson's algorithm for a uniform spanning tree
A demonstration of the growing percolation interface
Slides for Oded's talk at MSRI (May 2001)
Abstract: There are numerous predictions from statistical physics regarding random systems in the plane which were until recently beyond the reach of mathematical understanding. Some of the better known examples include percolation and the Ising model. We will focus on percolation and describe our growing understanding of it through a sequence of insights (which are simple in hindsight) from 1960 through today. |
Conformally Invariant Scaling Limits (ICM Madrid 2006
plenary lecture)
Abstract:
Many natural random processes on grids in the plane conjecturally
exhibit conformal invariance in the scaling limit. The archetypical
example is the simple random walk, whose scaling limit is Brownian
motion.
This talk will describe an explicit new conformally invariant random process, SLE(k), which depends on a parameter k > 0. For some values of the parameter k, it is conjectured that SLE(k) is equal to the scaling limit of fundamental random processes: SLE(2) is conjectured to be the scaling limit of loop-erased random walk, SLE(4) is conjectured to be the scaling limit of domino contours, SLE(6) is conjectured to be the scaling limit of percolation cluster boundaries, and SLE(8) is conjectured to be the scaling limit of the uniform spanning tree Peano contour. These processes will be described, and some of the evidence for the conjectures will be presented. The process SLE(k) is defined by specifying the conformal map to its complement: the conformal map is obtained by solving Loewner's differential equation with Brownian motion as the driving parameter. In joint work with Greg Lawler and Wendelin Werner, we use the SLE(6) process to obtain exact values for some Brownian motion exponents. |
The 2D Gaussian Free Field Interface (2005)
Abstract:
An instance of the Gaussian free field is a random function defined
in a domain in R^d. It is a generalization of Brownian motion to
the case where time is multi-dimensional, and it is useful for modeling
many different kinds of random surfaces. In two dimension, it exhibits
conformal invariance.
During the talk we will define the Gaussian free field and also the discrete Gaussian free field and describe joint work with Scott Sheffield identifying the scaling limit of the level lines of the discrete Gaussian free field in two dimensions (as the lattice mesh tends to zero). These limits are random curves having Hausdorff dimension 3/2 a.s. |
Random Planar Triangulations (2003)
some random triangulations: one (in dual form) two |
Quantum Gravity: the Mathematics of Random Metrics (2008)
Scaling Limits of Dynamical and Near-Critical
Percolation (2008)
Abstract:
Following recent progress in the understanding of critical percolation in
the plane, we proceed to study two closely related models: dynamical
percolation and near-critical percolation. In dynamical percolation the
bits defining the percolation configuration undergo independent Poisson
updates. For near-critical percolation, one can similarly start from the
critical configuration and perform random Poisson modifications, but in
this case the modifications are monotone.
We prove that for site percolation of the triangular lattice each of these has a scaling limit when time is scaled appropriately and establish several properties of these limits. They are rotationally invariant, their time evolution is Markov and the dynamical percolation scaling limit is ergodic. Though the scaling limits are not conformally invariant, we prove that they are conformally covariant in an appropriate sense. It is well-known that the minimal spanning tree is closely related to critical and near-critical percolation. We use this connection to deduce statements about a variant of the minimal spanning tree that is related to site percolation on the triangular lattice. For this variant, we prove, for example, that the minimal spanning tree scaling limit is rotationally invariant. Several of these results were conjectured by Camia, Fontes and Newman. Our proofs partially follow their heuristic narrative. Associated movies: site movie bond movie ensemble movie |
The Percolation Fourier Spectrum (2007)
Abstract: A real valued function f on the discrete cube {-1,1}^n has an expansion of the form f(x)=\sum_S F(S) u_S(x), where u_S(x) = \prod_{j\in S} x_j, and the sum is over all subsets S of the index set [n]. When f takes values in {-1,1}, the measure on such S given by mu(S)=F(S)^2 is a probability measure. The distribution of |S| under this measure encodes important aspects of the behaviour of f under noise and under the natural dynamics on the discrete cube. In joint work with Christophe Garban and Gabor Pete we describe mu in the case where f = 1 or -1, depending on whether there is a percolation crossing or not. We derive consequences for the behavior of percolation under noise and for exceptional times of dynamical percolation. Background in percolation theory will not be assumed. |