arXiv subjects

Nina Holden

Publications and source records attributed to Nina Holden.

At least 37 records · Page 2Linked to original sources

arXiv2021

Convergence of uniform triangulations under the Cardy embedding

We consider an embedding of planar maps into an equilateral triangle $Δ$ which we call the Cardy embedding. The embedding is a discrete approximation of a conformal map based on percolation observables that are used in Smirnov's proof of Cardy's formula. Under the Cardy embedding, the planar map induces a metric and an area measure on $Δ$ and a boundary measure on $\partial Δ$. We prove that for uniformly sampled triangulations, the metric and the measures converge jointly in the scaling limit to the Brownian disk conformally embedded into $Δ$ (i.e., to the $\sqrt{8/3}$-Liouville quantum gravity disk). As part of our proof, we prove scaling limit results for critical site percolation on the uniform triangulations, in a quenched sense. In particular, we establish the scaling limit of the percolation crossing probability for a uniformly sampled triangulation with four boundary marked points.

math.PR↗

arXiv2021

Natural parametrization of percolation interface and pivotal points

We prove that the interface of critical site percolation on the triangular lattice converges to SLE$_6$ in its natural parametrization, where the discrete interface is parametrized such that each edge is crossed in one unit of time, while the limiting curve is parametrized by the $7/4$-dimensional Minkowski content. We also prove that the scaling limit of counting measure on the pivotal points, which was proved to exist by Garban, Pete, and Schramm (2013), is the $3/4$-dimensional Minkowski content up to a deterministic multiplicative constant.

math.PR↗

arXiv2021

Minkowski content of Brownian cut points

Let $W(t)$, $0\leq t\leq T$, be a Brownian motion in $\mathbb{R}^d$, $d=2,3$. We say that $x$ is a cut point for $W$ if $x=W(t)$ for some $t\in(0,T)$ such that $W [0,t) $ and $W (t,T]$ are disjoint. In this work, we prove that a.s. the Minkowski content of the set of cut points for $W$ exists and is finite and non-trivial.

math.PR↗

arXiv2020

Scaling limit of triangulations of polygons

We prove that random triangulations of types I, II, and III with a simple boundary under the critical Boltzmann weight converge in the scaling limit to the Brownian disk. The proof uses a bijection due to Poulalhon and Schaeffer between type III triangulations of the $p$-gon and so-called blossoming forests. A variant of this bijection was also used by Addario-Berry and the first author to prove convergence of type III triangulations to the Brownian map, but new ideas are needed to handle the simple boundary. Our result is an ingredient in the program of the second and third authors on the convergence of uniform triangulations under the Cardy embedding.

math.PR↗

arXiv2020

Conformal welding for critical Liouville quantum gravity

Consider two critical Liouville quantum gravity surfaces (i.e., $γ$-LQG for $γ=2$), each with the topology of $\mathbb{H}$ and with infinite boundary length. We prove that there a.s. exists a conformal welding of the two surfaces, when the boundaries are identified according to quantum boundary length. This results in a critical LQG surface decorated by an independent SLE$_4$. Combined with the proof of uniqueness for such a welding, recently established by McEnteggart, Miller, and Qian (2018), this shows that the welding operation is well-defined. Our result is a critical analogue of Sheffield's quantum gravity zipper theorem (2016), which shows that a similar conformal welding for subcritical LQG (i.e., $γ$-LQG for $γ\in(0,2)$) is well-defined.

math.PR↗

arXiv2020

Subpolynomial trace reconstruction for random strings and arbitrary deletion probability

The insertion-deletion channel takes as input a bit string ${\bf x}\in\{0,1\}^{n}$, and outputs a string where bits have been deleted and inserted independently at random. The trace reconstruction problem is to recover $\bf x$ from many independent outputs (called "traces") of the insertion-deletion channel applied to $\bf x$. We show that if $\bf x$ is chosen uniformly at random, then $\exp(O(\log^{1/3} n))$ traces suffice to reconstruct $\bf x$ with high probability. For the deletion channel with deletion probability $q < 1/2$ the earlier upper bound was $\exp(O(\log^{1/2} n))$. The case of $q\geq 1/2$ or the case where insertions are allowed has not been previously analyzed, and therefore the earlier upper bound was as for worst-case strings, i.e., $\exp(O( n^{1/3}))$. We also show that our reconstruction algorithm runs in $n^{1+o(1)}$ time. A key ingredient in our proof is a delicate two-step alignment procedure where we estimate the location in each trace corresponding to a given bit of $\bf x$. The alignment is done by viewing the strings as random walks and comparing the increments in the walk associated with the input string and the trace, respectively.

math.PR↗

arXiv2020

A mating-of-trees approach for graph distances in random planar maps

We introduce a general technique for proving estimates for certain random planar maps which belong to the $γ$-Liouville quantum gravity (LQG) universality class for $γ\in (0,2)$. The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d.\ increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; $γ=\sqrt{8/3}$); and planar maps weighted by the number of different spanning trees ($γ=\sqrt 2$), bipolar orientations ($γ=\sqrt{4/3}$), or Schnyder woods ($γ=1$) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (1993) prediction for the Hausdorff dimension of $γ$-LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map $M$ to a mated-CRT map---a random planar map constructed from a correlated two-dimensional Brownian motion---using a strong coupling (Zaitsev, 1998) of the encoding walk for $M$ and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in $M$ from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when $γ=\sqrt{8/3}$, we instead deduce estimates for the $\sqrt{8/3}$-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.

math.PR↗

arXiv2019

Liouville quantum gravity with matter central charge in $(1,25)$: a probabilistic approach

There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge ${\mathbf{c}}_{\mathrm M}\in(-\infty,1]$. Via the DDK ansatz, LQG can equivalently be described as the random geometry obtained by exponentiating $γ$ times a variant of the planar Gaussian free field (GFF), where $γ\in(0,2]$ satisfies $\mathbf c_{\mathrm M}=25-6(2/γ+γ/2)^2$. Physics considerations suggest that LQG should also make sense in the regime when $\mathbf c_{\mathrm M}>1$. However, the behavior in this regime is rather mysterious in part because the corresponding value of $γ$ is complex, so analytic continuations of various formulas give complex answers which are difficult to interpret in a probabilistic setting. We introduce and study a discretization of LQG which makes sense for all values of $\mathbf c_{\mathrm M}\in(-\infty,25)$. Our discretization consists of a random planar map, defined as the adjacency graph of a tiling of the plane by dyadic squares which all have approximately the same "LQG size" with respect to the GFF. We prove that several formulas for dimension-related quantities are still valid for $\mathbf c_{\mathrm M}\in(1,25)$, with the caveat that the dimension is infinite when the formulas give a complex answer. In particular, we prove an extension of the (geometric) KPZ formula for $\mathbf c_{\mathrm M}\in(1,25)$, which gives a finite quantum dimension iff the Euclidean dimension is at most $(25-\mathbf c_{\mathrm M})/12$. We also show that the graph distance between typical points with respect to our discrete model grows polynomially whereas the cardinality of a graph distance ball of radius $r$ grows faster than any power of $r$ (which suggests that the Hausdorff dimension of LQG is infinite for $\mathbf c_{\mathrm M}\in(1,25)$). We include a substantial list of open problems.

math.PR↗

arXiv2019

Dimension transformation formula for conformal maps into the complement of an SLE curve

We prove a formula relating the Hausdorff dimension of a deterministic Borel subset of $\mathbb R$ and the Hausdorff dimension of its image under a conformal map from the upper half-plane to a complementary connected component of an SLE$_κ$ curve for $κ\not =4$. Our proof is based on the relationship between SLE and Liouville quantum gravity together with the one-dimensional KPZ formula of Rhodes-Vargas (2011) and the KPZ formula of Gwynne-Holden-Miller (2015). As an intermediate step we prove a KPZ formula which relates the Euclidean dimension of a subset of an SLE$_κ$ curve for $κ\in (0,4)\cup(4,8)$ and the dimension of the same set with respect to the $γ$-quantum natural parameterization of the curve induced by an independent Gaussian free field, $γ= \sqrt κ\wedge (4/\sqrtκ)$.

math.PR↗

arXiv2019

An almost sure KPZ relation for SLE and Brownian motion

The peanosphere construction of Duplantier, Miller, and Sheffield provides a means of representing a $γ$-Liouville quantum gravity (LQG) surface, $γ\in (0,2)$, decorated with a space-filling form of Schramm's SLE$_κ$, $κ= 16/γ^2 \in (4,\infty)$, $η$ as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion $Z$. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset $A$ of the range of $η$ which can be defined as a function of $η$ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $η^{-1}(A)$. This result serves to reduce the problem of computing the Hausdorff dimension of any set associated with an SLE, CLE, or related processes in the interior of a domain to the problem of computing the Hausdorff dimension of a certain set associated with a Brownian motion. For many natural examples, the associated Brownian motion set is well-known. As corollaries, we obtain new proofs of the Hausdorff dimensions of the SLE$_κ$ curve for $κ\not=4$; the double points and cut points of SLE$_κ$ for $κ>4$; and the intersection of two flow lines of a Gaussian free field. We also obtain the Hausdorff dimension of the set of $m$-tuple points of space-filling SLE$_κ$ for $κ>4$ and $m \geq 3$ by computing the Hausdorff dimension of the so-called $(m-2)$-tuple $π/2$-cone times of a correlated planar Brownian motion.

math.PR↗

arXiv2019

Lower bounds for trace reconstruction

In the trace reconstruction problem, an unknown bit string ${\bf x}\in\{0,1 \}^n$ is sent through a deletion channel where each bit is deleted independently with some probability $q\in(0,1)$, yielding a contracted string $\widetilde{\bf x}$. How many i.i.d.\ samples of $\widetilde{\bf x}$ are needed to reconstruct $\bf x$ with high probability? We prove that there exist ${\bf x},{\bf y} \in\{0,1 \}^n$ such that at least $c\, n^{5/4}/\sqrt{\log n}$ traces are required to distinguish between ${\bf x}$ and ${\bf y}$ for some absolute constant $c$, improving the previous lower bound of $c\,n$. Furthermore, our result improves the previously known lower bound for reconstruction of random strings from $c \log^2 n$ to $c \log^{9/4}n/\sqrt{\log \log n} $.

math.PR↗

arXiv2019

Liouville dynamical percolation

We construct and analyze a continuum dynamical percolation process which evolves in a random environment given by a $γ$-Liouville measure. The homogeneous counterpart of this process describes the scaling limit of discrete dynamical percolation on the rescaled triangular lattice. Our focus here is to study the same limiting dynamics, but where the speed of microscopic updates is highly inhomogeneous in space and is driven by the $γ$-Liouville measure associated with a two-dimensional log-correlated field $h$. Roughly speaking, this continuum percolation process evolves very rapidly where the field $h$ is high and barely moves where the field $h$ is low. Our main results can be summarized as follows. 1. First, we build this inhomogeneous dynamical percolation which we call $γ$-Liouville dynamical percolation (LDP) by taking the scaling limit of the associated process on the triangular lattice. We work with three different regimes each requiring different tools: $γ\in [0,2-\sqrt{5/2})$, $γ\in [2-\sqrt{5/2}, \sqrt{3/2})$, and $γ\in(\sqrt{3/2},2)$. 2. When $γ<\sqrt{3/2}$, we prove that $γ$-LDP is mixing in the Schramm-Smirnov space as $t\to \infty$, quenched in the log-correlated field $h$. On the contrary, when $γ>\sqrt{3/2}$ the process is frozen in time. The ergodicity result is a crucial piece of the Cardy embedding project of the second and fourth coauthors, where LDP for $γ=\sqrt{1/6}$ is used to study the scaling limit of a variant of dynamical percolation on uniform triangulations. 3. When $γ<\sqrt{3/4}$, we obtain quantitative bounds on the mixing of quad crossing events.

math.PR↗

arXiv2019

Gravitational allocation for uniform points on the sphere

Given a collection $\mathcal L$ of $n$ points on a sphere $\mathbf{S}^2_n$ of surface area $n$, a fair allocation is a partition of the sphere into $n$ parts each of area $1$, and each associated with a distinct point of $\mathcal L$. We show that if the $n$ points are chosen uniformly at random and the partition is defined by considering the gravitational field defined by the $n$ points, then the expected distance between a point on the sphere and the associated point of $\mathcal L$ is $O(\sqrt{\log n})$. We use our result to define a matching between two collections of $n$ independent and uniform points on the sphere, and prove that the expected distance between a pair of matched points is $O(\sqrt{\log n})$, which is optimal by a result of Ajtai, Komlós, and Tusnády.

math.PR↗

arXiv2019

Communication cost of consensus for nodes with limited memory

Motivated by applications in blockchains and sensor networks, we consider a model of $n$ nodes trying to reach consensus on their majority bit. Each node $i$ is assigned a bit at time zero, and is a finite automaton with $m$ bits of memory (i.e., $2^m$ states) and a Poisson clock. When the clock of $i$ rings, $i$ can choose to communicate, and is then matched to a uniformly chosen node $j$. The nodes $j$ and $i$ may update their states based on the state of the other node. Previous work has focused on minimizing the time to consensus and the probability of error, while our goal is minimizing the number of communications. We show that when $m>3 \log\log\log(n)$, consensus can be reached at linear communication cost, but this is impossible if $m<\log\log\log(n)$. We also study a synchronous variant of the model, where our upper and lower bounds on $m$ for achieving linear communication cost are $2\log\log\log(n)$ and $\log\log\log(n)$, respectively. A key step is to distinguish when nodes can become aware of knowing the majority bit and stop communicating. We show that this is impossible if their memory is too low.

cs.DC↗

arXiv2018

A distance exponent for Liouville quantum gravity

Let $γ\in (0,2)$ and let $h$ be the random distribution on $\mathbb C$ which describes a $γ$-Liouville quantum gravity (LQG) cone. Also let $κ= 16/γ^2 >4$ and let $η$ be a whole-plane space-filling SLE$_κ$ curve sampled independent from $h$ and parametrized by $γ$-quantum mass with respect to $h$. We study a family $\{\mathcal G^ε\}_{ε>0}$ of planar maps associated with $(h, η)$ called the \textit{LQG structure graphs} (a.k.a.\ \textit{mated-CRT maps}) which we conjecture converge in probability in the scaling limit with respect to the Gromov-Hausdorff topology to a random metric space associated with $γ$-LQG. In particular, $\mathcal G^ε$ is the graph whose vertex set is $ε\mathbb Z$, with two such vertices $x_1,x_2\in ε\mathbb Z$ connected by an edge if and only if the corresponding curve segments $η([x_1-ε, x_1])$ and $η([x_2-ε,x_2])$ share a non-trivial boundary arc. Due to the peanosphere description of SLE-decorated LQG due to Duplantier, Miller, and Sheffield (2014), the graph $\mathcal G^ε$ can equivalently be expressed as an explicit functional of a correlated two-dimensional Brownian motion, so can be studied without any reference to SLE or LQG. We prove non-trivial upper and lower bounds for the cardinality of a graph-distance ball of radius $n$ in $\mathcal G^ε$ which are consistent with the prediction of Watabiki (1993) for the Hausdorff dimension of LQG. Using subadditivity arguments, we also prove that there is an exponent $χ> 0$ for which the expected graph distance between generic points in the subgraph of $\mathcal G^ε$ corresponding to the segment $η([0,1])$ is of order $ε^{-χ+ o_ε(1)}$, and this distance is extremely unlikely to be larger than $ε^{-χ+ o_ε(1)}$.

math.PR↗

arXiv2018

Sparse exchangeable graphs and their limits via graphon processes

In a recent paper, Caron and Fox suggest a probabilistic model for sparse graphs which are exchangeable when associating each vertex with a time parameter in $\mathbb{R}_+$. Here we show that by generalizing the classical definition of graphons as functions over probability spaces to functions over $σ$-finite measure spaces, we can model a large family of exchangeable graphs, including the Caron-Fox graphs and the traditional exchangeable dense graphs as special cases. Explicitly, modelling the underlying space of features by a $σ$-finite measure space $(S,\mathcal{S},μ)$ and the connection probabilities by an integrable function $W\colon S\times S\to [0,1]$, we construct a random family $(G_t)_{t\geq 0}$ of growing graphs such that the vertices of $G_t$ are given by a Poisson point process on $S$ with intensity $tμ$, with two points $x,y$ of the point process connected with probability $W(x,y)$. We call such a random family a graphon process. We prove that a graphon process has convergent subgraph frequencies (with possibly infinite limits) and that, in the natural extension of the cut metric to our setting, the sequence converges to the generating graphon. We also show that the underlying graphon is identifiable only as an equivalence class over graphons with cut distance zero. More generally, we study metric convergence for arbitrary (not necessarily random) sequences of graphs, and show that a sequence of graphs has a convergent subsequence if and only if it has a subsequence satisfying a property we call uniform regularity of tails. Finally, we prove that every graphon is equivalent to a graphon on $\mathbb{R}_+$ equipped with Lebesgue measure.

math.PR↗

arXiv2018

Negative moments for Gaussian multiplicative chaos on fractal sets

The objective of this note is to study the probability that the total mass of a sub-critical Gaussian multiplicative chaos (GMC) with arbitrary base measure $σ$ is small. When $σ$ has some continuous density w.r.t Lebesgue measure, a scaling argument shows that the logarithm of the total GMC mass is sub-Gaussian near $-\infty$. However, when $σ$ has no scaling properties, the situation is much less clear. In this paper, we prove that for any base measure $σ$, the total GMC mass has negative moments of all orders.

math.PR↗

arXiv2018

SLE as a mating of trees in Euclidean geometry

The mating of trees approach to Schramm-Loewner evolution (SLE) in the random geometry of Liouville quantum gravity (LQG) has been recently developed by Duplantier-Miller-Sheffield (2014). In this paper we consider the mating of trees approach to SLE in Euclidean geometry. Let $η$ be a whole-plane space-filling SLE with parameter $κ>4$, parameterized by Lebesgue measure. The main observable in the mating of trees approach is the contour function, a two-dimensional continuous process describing the evolution of the Minkowski content of the left and right frontier of $η$. We prove regularity properties of the contour function and show that (as in the LQG case) it encodes all the information about the curve $η$. We also prove that the uniform spanning tree on $\mathbb Z^2$ converges to $\mathrm{SLE}_8$ in the natural topology associated with the mating of trees approach.

math.PR↗