arXiv subjects

Jason Miller

Publications and source records attributed to Jason Miller.

At least 73 records · Page 4Linked to original sources

arXiv2019

Harmonic functions on mated-CRT maps

A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar map models (e.g., uniform triangulations and spanning tree-weighted maps), and are closely related to $γ$-Liouville quantum gravity (LQG) for $γ\in (0,2)$ if we take the correlation to be $-\cos(πγ^2/4)$. We prove estimates for the Dirichlet energy and the modulus of continuity of a large class of discrete harmonic functions on mated-CRT maps, which provide a general toolbox for the study of the quantitative properties random walk and discrete conformal embeddings for these maps. For example, our results give an independent proof that the simple random walk on the mated-CRT map is recurrent, and a polynomial upper bound for the maximum length of the edges of the mated-CRT map under a version of the Tutte embedding. Our results are also used in other work by the first two authors which shows that for a class of random planar maps --- including mated-CRT maps and the UIPT --- the spectral dimension is two (i.e., the return probability of the simple random walk to its starting point after $n$ steps is $n^{-1+o_n(1)}$) and the typical exit time of the walk from a graph-distance ball is bounded below by the volume of the ball, up to a polylogarithmic factor.

math.PR↗

arXiv2019

Uniqueness of the welding problem for SLE and Liouville quantum gravity

We give a simple set of geometric conditions on curves $η$, $\tildeη$ in ${\mathbf H}$ from $0$ to $\infty$ so that if $φ\colon {\mathbf H} \to {\mathbf H}$ is a homeomorphism which is conformal off $η$ with $φ(η) = \tildeη$ then $φ$ is a conformal automorphism of ${\mathbf H}$. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if $η$ is a non-space-filling SLE$_κ$ curve in ${\mathbf H}$ from $0$ to $\infty$ and $φ$ is a homeomorphism which is conformal on ${\mathbf H} \setminus η$ and $φ(η)$, $η$ are equal in distribution then $φ$ is a conformal automorphism of ${\mathbf H}$. Applying this result for $κ=4$ establishes that the welding operation for critical ($γ=2$) Liouville quantum gravity (LQG) is well-defined. Applying it for $κ\in (4,8)$ gives a new proof that the welding of two independent $κ/4$-stable looptrees of quantum disks to produce an SLE$_κ$ on top of an independent $4/\sqrtκ$-LQG surface is well-defined.

math.PR↗

arXiv2018

Non-simple SLE curves are not determined by their range

We show that when observing the range of a chordal SLE$_κ$ curve for $κ\in (4,8)$, it is not possible to recover the order in which the points have been visited. We also derive related results about conformal loop ensembles (CLE): (i) The loops in a CLE$_κ$ for $κ\in (4,8)$ are not determined by the CLE$_κ$ gasket. (ii) The continuum percolation interfaces defined in the fractal carpets of conformal loop ensembles CLE$_κ$ for $κ\in (8/3, 4)$ (we defined these percolation interfaces in previous work, and showed there that they are SLE$_{16/κ}$ curves) are not determined by the CLE$_κ$ carpet that they are defined in.

math.PR↗

arXiv2018

Liouville quantum gravity spheres as matings of finite-diameter trees

We show that the unit area Liouville quantum gravity sphere can be constructed in two equivalent ways. The first, which was introduced by the authors and Duplantier, uses a Bessel excursion measure to produce a Gaussian free field variant on the cylinder. The second uses a correlated Brownian loop and a "mating of trees" to produce a Liouville quantum gravity sphere decorated by a space-filling path. In the special case that $γ=\sqrt{8/3}$, we present a third equivalent construction, which uses the excursion measure of a $3/2$-stable Lévy process (with only upward jumps) to produce a pair of trees of quantum disks that can be mated to produce a sphere decorated by SLE$_6$. This construction is relevant to a program for showing that the $γ=\sqrt{8/3}$ Liouville quantum gravity sphere is equivalent to the Brownian map.

math.PR↗

arXiv2018

Metric gluing of Brownian and $\sqrt{8/3}$-Liouville quantum gravity surfaces

In a recent series of works, Miller and Sheffield constructed a metric on $\sqrt{8/3}$-Liouville quantum gravity (LQG) under which $\sqrt{8/3}$-LQG surfaces (e.g., the LQG sphere, wedge, cone, and disk) are isometric to their Brownian surface counterparts (e.g., the Brownian map, half-plane, plane, and disk). We identify the metric gluings of certain collections of independent $\sqrt{8/3}$-LQG surfaces with boundaries identified together according to LQG length along their boundaries. Our results imply in particular that the metric gluing of two independent instances of the Brownian half-plane along their positive boundaries is isometric to a certain LQG wedge decorated by an independent chordal SLE$_{8/3}$ curve. If one identifies the two sides of the boundary of a single Brownian half-plane, one obtains a certain LQG cone decorated by an independent whole-plane SLE$_{8/3}$. If one identifies the entire boundaries of two Brownian half-planes, one obtains a different LQG cone and the interface between them is a two-sided variant of whole-plane SLE$_{8/3}$. Combined with another work of the authors, the present work identifies the scaling limit of self-avoiding walk on random quadrangulations with SLE$_{8/3}$ on $\sqrt{8/3}$-LQG.

math.PR↗

arXiv2018

Cut-off for lamplighter chains on tori: dimension interpolation and phase transition

Given a finite, connected graph $G$, the lamplighter chain on $G$ is the lazy random walk $X^\diamond$ on the associated lamplighter graph $G^\diamond={\mathbf Z}_2 \wr G$. The mixing time of the lamplighter chain on the torus ${\mathbf Z}_n^d$ is known to have a cutoff at a time asymptotic to the cover time of ${\mathbf Z}_n^d$ if $d=2$, and to half the cover time if $d \ge 3$. We show that the mixing time of the lamplighter chain on $G_n(a)={\mathbf Z}_n^2 \times {\mathbf Z}_{a \log n}$ has a cutoff at $ψ(a)$ times the cover time of $G_n(a)$ as $n \to \infty$, where $ψ$ is an explicit weakly decreasing map from $(0,\infty)$ onto $[1/2,1)$. In particular, as $a > 0$ varies, the threshold continuously interpolates between the known thresholds for ${\mathbf Z}_n^2$ and ${\mathbf Z}_n^3$. Perhaps surprisingly, we find a phase transition (non-smoothness of $ψ$) at the point $a_*=πr_3 (1+\sqrt{2})$, where high dimensional behavior ($ψ(a)=1/2$ for all $a \ge a_*$) commences. Here $r_3$ is the effective resistance from $0$ to $\infty$ in ${\mathbf Z}^3$.

math.PR↗

arXiv2018

Chordal SLE$_6$ explorations of a quantum disk

We consider a particular type of $\sqrt{8/3}$-Liouville quantum gravity surface called a doubly marked quantum disk (equivalently, a Brownian disk) decorated by an independent chordal SLE$_6$ curve $η$ between its marked boundary points. We obtain descriptions of the law of the quantum surfaces parameterized by the complementary connected components of $η([0,t])$ for each time $t \geq 0$ as well as the law of the left/right $\sqrt{8/3}$-quantum boundary length process for $η$.

math.PR↗

arXiv2018

Connection probabilities for conformal loop ensembles

The goal of the present paper is to explain, based on properties of the conformal loop ensembles CLE$_κ$ (both with simple and non-simple loops, i.e., for the whole range $κ\in (8/3, 8)$) how to derive the connection probabilities in conformal rectangles for a conditioned version of CLE$_κ$ which can be interpreted as a CLE$_κ$ with wired/free/wired/free boundary conditions on four boundary arcs (the wired parts being viewed as portions of to-be-completed loops). In particular, in the case of a conformal square, we prove that the probability that the two wired sides hook up so that they create one single loop is equal to $1/(1 - 2 \cos (4 π/ κ))$. Comparing this with the corresponding connection probabilities for discrete O(N) models for instance indicates that if a dilute O(N) model (respectively a critical FK(q)-percolation model on the square lattice) has a non-trivial conformally invariant scaling limit, then necessarily this scaling limit is CLE$_κ$ where $κ$ is the value in $(8/3, 4]$ such that $-2 \cos (4 π/ κ)$ is equal to $N$ (resp. the value in $[4,8)$ such that $-2 \cos (4π/ κ)$ is equal to $\sqrt {q}$). Our arguments and computations build on the one hand on Dubédat's SLE commutation relations (as developed and used by Dubédat, Zhan or Bauer-Bernard-Kytölä) and on the other hand, on the construction and properties of the conformal loop ensembles and their relation to Brownian loop-soups, restriction measures, and the Gaussian free field, as recently derived in works with Sheffield and with Qian.

math.PR↗

arXiv2018

Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk

We prove that the free Boltzmann quadrangulation with simple boundary and fixed perimeter, equipped with its graph metric, natural area measure, and the path which traces its boundary converges in the scaling limit to the free Boltzmann Brownian disk. The topology of convergence is the so-called Gromov-Hausdorff-Prokhorov-uniform (GHPU) topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces. From this we deduce that a random quadrangulation of the sphere decorated by a $2l$-step self-avoiding loop converges in law in the GHPU topology to the random curve-decorated metric measure space obtained by gluing together two independent Brownian disks along their boundaries.

math.PR↗

arXiv2017

Dimension of the SLE light cone, the SLE fan, and SLE$_κ(ρ)$ for $κ\in (0,4)$ and $ρ\in [\tfracκ{2}-4,-2)$

Suppose that $h$ is a Gaussian free field (GFF) on a planar domain. Fix $κ\in (0,4)$. The SLE$_κ$ light cone ${\mathbf L}(θ)$ of $h$ with opening angle $θ\in [0,π]$ is the set of points reachable from a given boundary point by angle-varying flow lines of the (formal) vector field $e^{i h/χ}$, $χ= \tfrac{2}{\sqrtκ} - \tfrac{\sqrtκ}{2}$, with angles in $[-\tfracθ{2},\tfracθ{2}]$. We derive the Hausdorff dimension of ${\mathbf L}(θ)$. If $θ=0$ then ${\mathbf L}(θ)$ is an ordinary SLE$_κ$ curve (with $κ< 4$); if $θ= π$ then ${\mathbf L}(θ)$ is the range of an SLE$_{κ'}$ curve ($κ' = 16/κ> 4$). In these extremes, this leads to a new proof of the Hausdorff dimension formula for SLE. We also consider SLE$_κ(ρ)$ processes, which were originally only defined for $ρ> -2$, but which can also be defined for $ρ\leq -2$ using Lévy compensation. The range of an SLE$_κ(ρ)$ is qualitatively different when $ρ\leq -2$. In particular, these curves are self-intersecting for $κ< 4$ and double points are dense, while ordinary SLE$_κ$ is simple. It was previously shown (Miller-Sheffield, 2016) that certain SLE$_κ(ρ)$ curves agree in law with certain light cones. Combining this with other known results, we obtain a general formula for the Hausdorff dimension of SLE$_κ(ρ)$ for all values of $ρ$. Finally, we show that the Hausdorff dimension of the so-called SLE$_κ$ fan is the same as that of ordinary SLE$_κ$.

math.PR↗

arXiv2017

Active spanning trees with bending energy on planar maps and SLE-decorated Liouville quantum gravity for $κ> 8$

We introduce a two-parameter family of probability measures on spanning trees of a planar map. One of the parameters controls the activity of the spanning tree and the other is a measure of its bending energy. When the bending parameter is 1, we recover the active spanning tree model, which is closely related to the critical Fortuin--Kasteleyn model. A random planar map decorated by a spanning tree sampled from our model can be encoded by means of a generalized version of Sheffield's hamburger-cheeseburger bijection. Using this encoding, we prove that for a range of parameter values (including the ones corresponding to maps decorated by an active spanning tree), the infinite-volume limit of spanning-tree-decorated planar maps sampled from our model converges in the peanosphere sense, upon rescaling, to an SLE$_κ$-decorated $γ$-Liouville quantum cone with $κ> 8$ and $γ= 4/\sqrtκ\in (0,\sqrt 2)$.

math.PR↗

arXiv2017

Liouville quantum gravity as a metric space and a scaling limit

Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has its roots in string theory and conformal field theory from the 1980s and 1990s. The second is the Brownian map, which has its roots in planar map combinatorics from the 1960s together with recent scaling limit results. This article surveys a series of works with Sheffield in which it is shown that Liouville quantum gravity (LQG) with parameter $γ=\sqrt{8/3}$ is equivalent to the Brownian map. We also briefly describe a series of works with Gwynne which use the $\sqrt{8/3}$-LQG metric to prove the convergence of self-avoiding walks and percolation on random planar maps towards SLE$_{8/3}$ and SLE$_6$, respectively, on a Brownian surface.

math.PR↗

arXiv2017

Almost sure multifractal spectrum of SLE

Suppose that $η$ is a Schramm-Loewner evolution (SLE$_κ$) in a smoothly bounded simply connected domain $D \subset {\mathbb C}$ and that $ϕ$ is a conformal map from ${\mathbb D}$ to a connected component of $D \setminus η([0,t])$ for some $t>0$. The multifractal spectrum of $η$ is the function $(-1,1) \to [0,\infty)$ which, for each $s \in (-1,1)$, gives the Hausdorff dimension of the set of points $x \in \partial {\mathbb D}$ such that $|ϕ'( (1-ε) x)| = ε^{-s+o(1)}$ as $ε\to 0$. We rigorously compute the a.s. multifractal spectrum of SLE, confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the a.s. bulk integral means spectrum of SLE and we obtain a new derivation of the a.s. Hausdorff dimension of the SLE curve for $κ\leq 4$. Our results also hold for the SLE$_κ(\underline ρ)$ processes with general vectors of weight $\underlineρ$.

math.PR↗

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

CLE percolations

Conformal loop ensembles are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any CLE loop is a natural random and conformally invariant analog of the Sierpinski gasket or carpet. In the present paper, we derive a direct relationship between each CLE consisting of simple disjoint loops (CLE($κ$) with $κ$ between 8/3 and 4) and the corresponding CLE($κ'$) where $κ':=16/κ$, a CLE consisting of non-disjoint loops. This is the continuum analog of the Edwards-Sokal coupling (between the q-state Potts model and the associated FK random cluster model) and its generalization to non-integer q. Like its discrete analog, our continuum correspondence has two directions. First, we show that one can construct (variants of) CLE($κ$) as follows: sample a CLE($κ'$), then use a biased coin to independently color each loop one of two colors, and then consider the outer boundaries of the clusters of loops of a given color. Second, we show how to interpret CLE($κ'$) loops as interfaces of a continuum analog of critical percolation within a CLE($κ)$ carpet. This is the first description of continuous percolation interfaces in fractal domains. These constructions provide new interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain results about generalized SLE$(κ;ρ)$ curves, and define a continuous family of natural CLE variants called boundary conformal loop ensembles (BCLEs) that share some (but not all) of the conformal symmetries that characterize CLEs, and that should be scaling limits of critical models with special boundary conditions. We extend the CLE correspondence to a BCLE correspondence that makes sense for all $κ$ between 2 and 4.

math.PR↗

arXiv2017

The six-vertex model and Schramm-Loewner evolution

Square ice is a statistical mechanics model for two-dimensional ice, widely believed to have a conformally invariant scaling limit. We associate a Peano (space filling) curve to a square ice configuration, and more generally to a so-called 6-vertex model configuration, and argue that its scaling limit is a space-filling version of the random fractal curve SLE$_κ$, Schramm--Loewner evolution with parameter $κ$, where $4<κ\leq 12+8\sqrt{2}$. For square ice, $κ=12$. At the "free-fermion point" of the 6-vertex model, $κ=8+4\sqrt{3}$. These unusual values lie outside the classical interval $2\le κ\le 8$.

cond-mat.stat-mech↗

arXiv2017

Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees

We establish existence and uniqueness for Gaussian free field flow lines started at {\em interior} points of a planar domain. We interpret these as rays of a random geometry with imaginary curvature and describe the way distinct rays intersect each other and the boundary. Previous works in this series treat rays started at {\em boundary} points and use Gaussian free field machinery to determine which chordal \SLE_κ(ρ_1; ρ_2) processes are time-reversible when κ< 8. Here we extend these results to whole-plane \SLE_κ(ρ) and establish continuity and transience of these paths. In particular, we extend ordinary whole-plane SLE reversibility (established by Zhan for κ\in [0,4]) to all κ\in [0,8]. We also show that the rays of a given angle (with variable starting point) form a space-filling planar tree. Each branch is a form of SLE_κfor some κ\in (0, 4), and the curve that traces the tree in the natural order (hitting x before y if the branch from x is left of the branch from y) is a space-filling form of \SLE_{κ'} where κ':= 16/κ\in (4, \infty). By varying the boundary data we obtain, for each κ'>4, a family of space-filling variants of \SLE_{κ'}(ρ) whose time reversals belong to the same family. When κ' \geq 8, ordinary \SLE_{κ'} belongs to this family, and our result shows that its time-reversal is \SLE_{κ'}(κ'/2 - 4; κ'/2 - 4). As applications of this theory, we obtain the local finiteness of \CLE_{κ'}, for κ' \in (4,8), and describe the laws of the boundaries of \SLE_{κ'} processes stopped at stopping times.

math.PR↗

arXiv2016

Bipolar orientations on planar maps and SLE$_{12}$

We give bijections between bipolar-oriented (acyclic with unique source and sink) planar maps and certain random walks, which show that the uniformly random bipolar-oriented planar map, decorated by the "peano curve" surrounding the tree of left-most paths to the sink, converges in law with respect to the peanosphere topology to a $\sqrt{4/3}$-Liouville quantum gravity surface decorated by an independent Schramm-Loewner evolution with parameter $κ=12$ (i.e., SLE$_{12}$). This result is universal in the sense that it holds for bipolar-oriented triangulations, quadrangulations, $k$-angulations, and maps in which face sizes are mixed.

math.PR↗