Showing papers by "Nina Holden published in 2020"

An almost sure KPZ relation for SLE and Brownian motion

Abstract: The peanosphere construction of Duplantier, Miller and Sheffield provides a means of representing a $\gamma $-Liouville quantum gravity (LQG) surface, $\gamma \in (0,2)$, decorated with a space-filling form of Schramm’s $\mathrm{SLE}_{\kappa }$, $\kappa =16/\gamma^{2}\in (4,\infty)$, $\eta $ 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 $\eta $, which can be defined as a function of $\eta $ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $\eta^{-1}(A)$. This result serves to reduce the problem of computing the Hausdorff dimension of any set associated with an $\mathrm{SLE}$, $\mathrm{CLE}$ 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 $\mathrm{SLE}_{\kappa}$ curve for $\kappa eq4$; the double points and cut points of $\mathrm{SLE}_{\kappa }$ for $\kappa >4$; and the intersection of two flow lines of a Gaussian free field. We obtain the Hausdorff dimension of the set of $m$-tuple points of space-filling $\mathrm{SLE}_{\kappa }$ for $\kappa >4$ and $m\geq 3$ by computing the Hausdorff dimension of the so-called $(m-2)$-tuple $\pi /2$-cone times of a correlated planar Brownian motion.

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

Abstract: We introduce a general technique for proving estimates for certain random planar maps which belong to the $$\gamma $$ -Liouville quantum gravity (LQG) universality class for $$\gamma \in (0,2)$$ . 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; $$\gamma =\sqrt{8/3}$$ ); and planar maps weighted by the number of different spanning trees ( $$\gamma =\sqrt{2}$$ ), bipolar orientations ( $$\gamma =\sqrt{4/3}$$ ), or Schnyder woods ( $$\gamma =1$$ ) 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 (Prog Theor Phys Suppl 114:1–17, 1993) prediction for the Hausdorff dimension of $$\gamma $$ -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 in ESAIM Probab Stat 2:41–108, 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 $$\gamma =\sqrt{8/3}$$ , 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.

Lower bounds for trace reconstruction

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

Liouville Quantum Gravity with Matter Central Charge in (1, 25): A Probabilistic Approach

Abstract: 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 $$\gamma $$ times a variant of the planar Gaussian free field, where $$\gamma \in (0,2]$$ satisfies $${\mathbf {c}}_{\mathrm M} = 25 - 6(2/\gamma + \gamma /2)^2$$ . 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 $$\gamma $$ 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 Gaussian free field. 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 if and only if 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 in the case when $${\mathbf {c}}_{\mathrm M} \in (1,25)$$ is infinite). We include a substantial list of open problems.

Conformal welding of quantum disks

Abstract: Two-pointed quantum disks are a family of finite-area random surfaces that arise naturally in Liouville quantum gravity. In this paper we show that conformally welding two quantum disks according to their boundary lengths gives another quantum disk decorated with a chordal SLE curve. This extends the classical result of Sheffield (2010) and Duplantier-Miller-Sheffield (2014) on the welding of infinite-area two-pointed quantum surfaces, which is fundamental to the mating-of-trees theory. Our results can be used to give a systematic treatment of mating-of-trees for finite-area quantum surfaces. Moreover, it serves as a key ingredient of the forthcoming work of the authors and G. Remy relating Liouville conformal field theory and SLE.

Scaling limit of triangulations of polygons

Abstract: 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.

Communication cost of consensus for nodes with limited memory

Abstract: Motivated by applications in wireless networks and the Internet of Things, we consider a model of n nodes trying to reach consensus with high probability on their majority bit. Each node i is assigned a bit at time 0 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 with linear communication cost, but this is impossible if m log ⁡ log ⁡ log ( n ) . 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.

Scaling limits of the Schelling model

Abstract: The Schelling model of segregation, introduced by Schelling in 1969 as a model for residential segregation in cities, describes how populations of multiple types self-organize to form homogeneous clusters of one type. In this model, vertices in an N-dimensional lattice are initially assigned types randomly. As time evolves, the type at a vertex v has a tendency to be replaced with the most common type within distance w of v. We present the first mathematical description of the dynamical scaling limit of this model as w tends to infinity and the lattice is correspondingly rescaled. We do this by deriving an integro-differential equation for the limiting Schelling dynamics and proving almost sure existence and uniqueness of the solutions when the initial conditions are described by white noise. The evolving fields are in some sense very “rough” but we are able to make rigorous sense of the evolution. In a key lemma, we show that for certain Gaussian fields h, the supremum of the occupation density of $$h-\phi $$ at zero (taken over all 1-Lipschitz functions $$\phi $$) is almost surely finite, thereby extending a result of Bass and Burdzy. In the one dimensional case, we also describe the scaling limit of the limiting clusters obtained at time infinity, thereby resolving a conjecture of Brandt, Immorlica, Kamath, and Kleinberg.

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

Abstract: 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 $$\hbox {SLE}_\kappa $$ curve for $$\kappa ot =4$$. Our proof is based on the relationship between SLE and Liouville quantum gravity together with the one-dimensional KPZ formula of Rhodes and Vargas (ESAIM Probab Stat 15:358–371, 2011) and the KPZ formula of Gwynne et al. (Ann Probab, 2015). As an intermediate step we prove a KPZ formula which relates the Euclidean dimension of a subset of an $$\hbox {SLE}_\kappa $$ curve for $$\kappa \in (0,4)\cup (4,8)$$ and the dimension of the same set with respect to the $$\gamma $$-quantum natural parameterization of the curve induced by an independent Gaussian free field, $$\gamma = \sqrt{\kappa }\wedge (4/\sqrt{\kappa })$$.