arXiv subjects

Ewain Gwynne

Publications and source records attributed to Ewain Gwynne.

79 records · Page 5Linked to original sources

arXiv2017

Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov-Hausdorff-Prokhorov-uniform topology

We prove that the uniform infinite half-plane quadrangulation (UIHPQ), with either general or simple boundary, equipped with its graph distance, its natural area measure, and the curve which traces its boundary, converges in the scaling limit to the Brownian half-plane. The topology of convergence is given by the so-called Gromov-Hausdorff-Prokhorov-uniform (GHPU) metric on curve-decorated metric measure spaces, which is a generalization of the Gromov-Hausdorff metric whereby two such spaces $(X_1, d_1 , μ_1,η_1)$ and $(X_2, d_2 , μ_2,η_2)$ are close if they can be isometrically embedded into a common metric space in such a way that the spaces $X_1$ and $X_2$ are close in the Hausdorff distance, the measures $μ_1$ and $μ_2$ are close in the Prokhorov distance, and the curves $η_1$ and $η_2$ are close in the uniform distance.

math.PR↗

arXiv2017

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

We continue our study of the inventory accumulation introduced by Sheffield (2011), which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. We prove various \emph{local estimates} for the inventory accumulation model, i.e., estimates for the precise number of symbols of a given type in a reduced word sampled from the model. Using our estimates, we obtain the scaling limit of the associated two-dimensional random walk conditioned on the event that it stays in the first quadrant for one unit of time and ends up at a particular position in the interior of the first quadrant. We also obtain the exponent for the probability that a word of length $2n$ sampled from the inventory accumulation model corresponds to an empty reduced word, which is equivalent to an asymptotic formula for the partition function of the critical FK planar map model. The estimates of this paper will be used in a subsequent paper to obtain the scaling limit of the lattice walk associated with a finite-volume FK planar map.

math.PR↗

arXiv2016

Asymptotic behavior of the Eden model with positively homogeneous edge weights

Let $d\in\mathbb N$, $α\in\mathbb R$, and let $f :\mathbb R^d\setminus \{0\} \rightarrow (0,\infty)$ be locally Lipschitz and positively homogeneous of degree $α$ (e.g. $f$ could be the $α$th power of a norm on $\mathbb R^d$). We study a generalization of the Eden model on $\mathbb Z^d$ wherein the next edge added to the cluster is chosen from the set of all edges incident to the current cluster with probability proportional to the value of $f$ at the midpoint of this edge, rather than uniformly. This model is equivalent to a variant of first passage percolation where the edge passage times are independent exponential random variables with parameters given by the value of $f$ at the midpoint of the edge. We prove that the $f$-weighted Eden model clusters have an a.s. deterministic limit shape if $α< 1$, which is an explicit functional of $f$ and the limit shape of the standard Eden model, and estimate the rate of convergence to this limit shape. We also prove that if $α>1$, then there is a norm $ν$ on $\mathbb R^d$ (depending on $α$) such that if we set $f(z) = ν(z)^{ α}$, then the $f$-weighted Eden model clusters are a.s.\ contained in a Euclidean cone with opening angle $<π$ for all time. We further show that there does \emph{not} exist a norm on $\mathbb R^d$ for which this latter statement holds for all $α>1$; and that there is no choice of function $f$ for which the above statement holds with $α=1$. Our basic approach is to compare the local behavior of the $f$-weighted first passage percolation to that of unweighted first passage percolation with iid exponential edge weights (which is equivalent to the unweighted Eden model). We include a list of open problems and several computer simulations.

math.PR↗

arXiv2016

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

Sheffield (2011) introduced an inventory accumulation model which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. He showed that a certain two-dimensional random walk associated with an infinite-volume version of the model converges in the scaling limit to a correlated planar Brownian motion. We improve on this scaling limit result by showing that the times corresponding to FK loops (or "flexible orders") in the inventory accumulation model converge in the scaling limit to the $π/2$-cone times of the correlated Brownian motion. This statement implies a scaling limit result for the joint law of the areas and boundary lengths of the bounded complementary connected components of the FK loops on the infinite-volume planar map. In light of the encoding of Duplantier, Miller, and Sheffield (2014), the limiting object coincides with the joint law of the areas and boundary lengths of the bounded complementary connected components of a collection of CLE$_κ$ loops on an independent Liouville quantum gravity cone.

math.PR↗

arXiv2016

Brownian motion correlation in the peanosphere for $κ> 8$

The peanosphere (or "mating of trees") construction of Duplantier, Miller, and Sheffield encodes certain types of $γ$-Liouville quantum gravity (LQG) surfaces ($γ\in (0,2)$) decorated with an independent SLE$_κ$ ($κ= 16/γ^2 > 4$) in terms of a correlated two-dimensional Brownian motion and provides a framework for showing that random planar maps decorated with statistical physics models converge to LQG decorated with an SLE. Previously, the correlation for the Brownian motion was only explicitly identified as $-\cos(4π/κ)$ for $κ\in (4,8]$ and unknown for $κ> 8$. The main result of this work is that this formula holds for all $κ> 4$. This supplies the missing ingredient for proving convergence results of the aforementioned type for $κ> 8$. Our proof is based on the calculation of a certain tail exponent for SLE$_κ$ on a quantum wedge and then matching it with an exponent which is well-known for Brownian motion.

math.PR↗

arXiv2015

Scaling limits for the critical Fortuin-Kastelyn model on a random planar map III: finite volume case

We prove scaling limit results for the finite-volume version of the inventory accumulation model of Sheffield (2011), which encodes a random planar map decorated by a collection of loops sampled from the critical Fortuin-Kasteleyn (FK) model. In particular, we prove that the random walk associated with the finite-volume version of this model converges in the scaling limit to a correlated Brownian motion $\dot Z$ conditioned to stay in the first quadrant for two units of time and satisfy $\dot Z(2) = 0$. We also show that the times which describe complementary connected components of FK loops in the discrete model converge to the $π/2$-cone times of $\dot Z$. Combined with recent results of Duplantier, Miller, and Sheffield, our results imply that many interesting functionals of the FK loops on a finite-volume FK planar map (e.g. their boundary lengths and areas) converge in the scaling limit to the corresponding "quantum" functionals of the CLE$_κ$ loops on a $4/\sqrtκ$-Liouville quantum gravity sphere for $κ\in (4,8)$. Our results are finite-volume analogues of the scaling limit theorems for the infinite-volume version of the inventory accumulation model proven by Sheffield (2011) and Gwynne, Mao, and Sun (2015).

math.PR↗

arXiv2011

On a Quaternionic Analogue of the Cross-Ratio

In this article we study an exact analogue of the cross-ratio for the algebra of quaternions H and use it to derive several interesting properties of quaternionic fractional linear transformations. In particular, we show that there exists a fractional linear transformation T on H mapping four distinct quaternions q_1, q_2, q_3 and q_4 into q'_1, q'_2, q'_3 and q'_4 respectively if and only if the quadruples (q_1, q_2, q_3, q_4) and (q'_1, q'_2, q'_3, q'_4) have the same cross-ratio. If such a fractional linear transformation T exists it is never unique. However, we prove that a fractional linear transformation on H is uniquely determined by specifying its values at five points in general position. We also prove some properties of the cross-ratio including criteria for four quaternions to lie on a single circle (or a line) and for five quaternions to lie on a single 2-sphere (or a 2-plane). As an application of the cross-ratio, we prove that fractional linear transformations on H map spheres (or affine subspaces) of dimension 1, 2 and 3 into spheres (or affine subspaces) of the same dimension.

math.CV↗