In this note, I will describe some highlights of Oded Schramm’s work in circle packings and the Koebe conjecture, as well as on SLE.

Oded Schramm (1961–2008) influenced greatly the development of percolation theory beyond the usual ℤ^d setting; in particular, the case of nonamenable lattices. Here, we review some of his work in this field.

We survey in this paper the main contributions of Oded Schramm related to noise sensitivity. We will describe in particular his various works which focused on the “spectral analysis” of critical percolation (and more generally of Boolean functions), his work on the shape-fluctuations of first passage percolation and finally his contributions to the model of dynamical percolation.

We prove Tsirelson’s conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also suggest a natural construction for the scaling limit of planar percolation, and more generally of any discrete planar model describing connectivity properties.

Let $\mathcal{S}_{n}$ be the permutation group on n elements, and consider a random walk on $\mathcal{S}_{n}$ whose step distribution is uniform on k-cycles. We prove a well-known conjecture that the mixing time of this process is (1/k)n log n, with threshold of width linear in n. Our proofs are elementary and purely probabilistic, and do not appeal to the representation theory of $\mathcal{S}_{n}$.

Gérard Watts predicted a formula for the probability in percolation that there is both a left–right and an up–down crossing, which was later proved by Julien Dubédat. Here we present a simpler proof due to Oded Schramm, which builds on Cardy’s formula in a conceptually appealing way: the triple derivative of Cardy’s formula is the sum of two multi-arm densities. The relative sizes of the two terms are computed with Girsanov conditioning. The triple integral of one of the terms is equivalent to Watts’ formula. For the relevant calculations, we present and annotate Schramm’s original (and remarkably elegant) Mathematica code.

We show that critical percolation on a product of two regular trees of degree ≥ 3 satisfies the triangle condition. The proof does not examine the degrees of vertices and is not “perturbative” in any sense. It relies on an unpublished lemma of Oded Schramm.

The Schramm–Loewner evolution (SLE_κ) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When κ < 8, an instance of SLE_κ is a random planar curve with almost sure Hausdorff dimension d = 1 + κ/8 < 2. This curve is conventionally parametrized by its half plane capacity, rather than by any measure of its d-dimensional volume.

For κ<8, we use a Doob–Meyer decomposition to construct the unique (under mild assumptions) Markovian parametrization of SLE_κ that transforms like a d-dimensional volume measure under conformal maps. We prove that this parametrization is nontrivial (i.e., the curve is not entirely traversed in zero time) for $\kappa< 4(7-\sqrt{33})=5.021\ldots$.

Given a homogeneous Poisson process on ℝ^d with intensity λ, we prove that it is possible to partition the points into two sets, as a deterministic function of the process, and in an isometry-equivariant way, so that each set of points forms a homogeneous Poisson process, with any given pair of intensities summing to λ. In particular, this answers a question of Ball [Electron. Commun. Probab. 10 (2005) 60–69], who proved that in d = 1, the Poisson points may be similarly partitioned (via a translation-equivariant function) so that one set forms a Poisson process of lower intensity, and asked whether the same is possible for all d. We do not know whether it is possible similarly to add points (again chosen as a deterministic function of a Poisson process) to obtain a Poisson process of higher intensity, but we prove that this is not possible under an additional finitariness condition.

The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial interpretation of their determinants in terms of so-called cycle rooted spanning forests (CRSFs). We construct natural measures on CRSFs for which the edges form a determinantal process.

This theory gives a natural generalization of the spanning tree process adapted to graphs embedded on surfaces. We give a number of other applications, for example, we compute the probability that a loop-erased random walk on a planar graph between two vertices on the outer boundary passes left of two given faces. This probability cannot be computed using the standard Laplacian alone.

In this paper, we consider a random walk and a random color scenery on ℤ. The increments of the walk and the colors of the scenery are assumed to be i.i.d. and to be independent of each other. We are interested in the random process of colors seen by the walk in the course of time. Bad configurations for this random process are the discontinuity points of the conditional probability distribution for the color seen at time zero given the colors seen at all later times.

We focus on the case where the random walk has increments 0, +1 or −1 with probability ε, (1 − ε)p and (1 − ε)(1 − p), respectively, with p ∈ [½, 1] and ε ∈ [0, 1), and where the scenery assigns the color black or white to the sites of ℤ with probability ½ each. We show that, remarkably, the set of bad configurations exhibits a crossover: for ε = 0 and p ∈ (½, ⅘) all configurations are bad, while for (p, ε) in an open neighborhood of (1, 0) all configurations are good. In addition, we show that for ε = 0 and p = ½ both bad and good configurations exist. We conjecture that for all ε ∈ [0, 1) the crossover value is unique and equals ⅘. Finally, we suggest an approach to handle the seemingly more difficult case where ε > 0 and p ∈ [½, ⅘), which will be pursued in future work.