The Rohde--Schramm theorem states that Schramm--Loewner Evolution with parameter $kappa$ (or SLE$_kappa$ for short) exists as a random curve, almost surely, if $kappa eq 8$. Here we give a new and concise proof of the result, based on the Liouville
quantum gravity coupling (or reverse coupling) with a Gaussian free field. This transforms the problem of estimating the derivative of the Loewner flow into estimating certain correlated Gaussian free fields. While the correlation between these fields is not easy to understand, a surprisingly simple argument allows us to recover a derivative exponent first obtained by Rohde and Schramm, subsequently shown to be optimal by Lawler and Viklund, which then implies the Rohde--Schramm theorem.
We study random walks on the giant component of the ErdH{o}s-Renyi random graph ${cal G}(n,p)$ where $p=lambda/n$ for $lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Ko
zma and Wormald, to have order $log^2 n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(log n)$ and concentrates it (the cutoff phenomenon occurs): the typical mixing is at $( u {bf d})^{-1}log n pm (log n)^{1/2+o(1)}$, where $ u$ and ${bf d}$ are the speed of random walk and dimension of harmonic measure on a ${rm Poisson}(lambda)$-Galton-Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the non-backtracking random walk.
We prove a conjecture raised by the work of Diaconis and Shahshahani (1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and fragmentation proce
sses and control the Kantorovitch distance by using a variant of a coupling due to Oded Schramm. Recasting our proof in the language of Ricci curvature, our proof establishes the occurrence of a phase transition, which takes the following form in the case of random transpositions: at time $cn/2$, the curvature is asymptotically zero for $cle 1$ and is strictly positive for $c>1$.
We introduce particle systems in one or more dimensions in which particles perform branching Brownian motion and the population size is kept constant equal to $N > 1$, through the following selection mechanism: at all times only the $N$ fittest parti
cles survive, while all the other particles are removed. Fitness is measured with respect to some given score function $s:R^d to R$. For some choices of the function $s$, it is proved that the cloud of particles travels at positive speed in some possibly random direction. In the case where $s$ is linear, we show under some assumptions on the initial configuration that the shape of the cloud scales like $log N$ in the direction parallel to motion but at least $c(log N)^{3/2}$ in the orthogonal direction for some $c > 0$. We conjecture that the exponent 3/2 is sharp. This result is equivalent to the following result of independent interest: in one-dimensional systems, the genealogical time is greater than $c(log N)^3$, thereby contributing a step towards the original predictions of Brunet and Derrida. We discuss several open problems and also explain how our results can be viewed as a rigorous justification of Weismanns arguments for the role of recombination in population genetics.
We introduce a Gibbs measure on nearest-neighbour paths of length $t$ in the Euclidean $d$-dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature $beta$. We prove that, for al
l $beta>0$, the random walk condensates to a set of diameter $(t/beta)^{1/3}$ in dimension $d=2$, up to a multiplicative constant. In all dimensions $dge 3$, we also prove that the volume is bounded above by $(t/beta)^{d/(d+1)}$ and the diameter is bounded below by $(t/beta)^{1/(d+1)}$. Similar results hold for a random walk conditioned to have local time greater than $beta$ everywhere in its range when $beta$ is larger than some explicit constant, which in dimension two is the logarithm of the connective constant.
We consider critical branching Brownian motion with absorption, in which there is initially a single particle at $x > 0$, particles move according to independent one-dimensional Brownian motions with the critical drift of $-sqrt{2}$, and particles ar
e absorbed when they reach zero. Here we obtain asymptotic results concerning the behavior of the process before the extinction time, as the position $x$ of the initial particle tends to infinity. We estimate the number of particles in the system at a given time and the position of the right-most particle. We also obtain asymptotic results for the configuration of particles at a typical time.
We consider branching Brownian motion on the real line with absorption at zero, in which particles move according to independent Brownian motions with the critical drift of $-sqrt{2}$. Kesten (1978) showed that almost surely this process eventually d
ies out. Here we obtain upper and lower bounds on the probability that the process survives until some large time $t$. These bounds improve upon results of Kesten (1978), and partially confirm nonrigorous predictions of Derrida and Simon (2007).
We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than $eps$, agrees up to generation $K$ with a regular $mu$-ary tree, where $mu$ is the essential minimum of the offspring distribution and the random vari
able $K$ is strongly concentrated near an explicit deterministic function growing like a multiple of $log(1/eps)$. More precisely, we show that if $muge 2$ then with high probability as $eps downarrow 0$, $K$ takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular $mu$-ary tree, providing an example of entropic repulsion where the limit has vanishing entropy.
