Dimensions of Two-Valued Sets via Imaginary Chaos

2020 ◽  
Author(s):  
Lukas Schoug ◽  
Avelio Sepúlveda ◽  
Fredrik Viklund
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Free Field ◽  
Point Estimate ◽  
Gaussian Free Field ◽  
Minkowski Content ◽  
The Real ◽  
Dimensional Measure

Abstract Two-valued sets are local sets of the 2D Gaussian free field (GFF) that can be thought of as representing all points of the domain that may be connected to the boundary by a curve on which the GFF takes values only in $[-a,b]$. Two-valued sets exist whenever $a+b\geq 2\lambda ,$ where $\lambda$ depends explicitly on the normalization of the GFF. We prove that the almost sure Hausdorff dimension of the two-valued set ${\mathbb{A}}_{-a,b}$ equals $d=2-2\lambda ^2/(a+b)^2$. For the key two-point estimate needed to give the lower bound on dimension, we use the real part of a “vertex field” built from the purely imaginary Gaussian multiplicative chaos. We also construct a non-trivial $d$-dimensional measure supported on ${\mathbb{A}}_{-a,b}$ and discuss its relation with the $d$-dimensional conformal Minkowski content of ${\mathbb{A}}_{-a,b}$.

scholarly journals The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

2020 ◽  
Vol 378 (1) ◽  
pp. 625-689 ◽  
Author(s):  
Ewain Gwynne
Keyword(s):  
Hausdorff Dimension ◽  
Quantum Gravity ◽  
Free Field ◽  
Gaussian Free Field ◽  
Hausdorff Dimensions ◽  
Thick Points

Abstract Let $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , let h be the planar Gaussian free field, and consider the $$\gamma $$ γ -Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a $$\gamma $$ γ -LQG metric ball with respect to the Euclidean (resp. $$\gamma $$ γ -LQG) metric is $$2 - \frac{\gamma }{d_\gamma }\left( \frac{2}{\gamma } + \frac{\gamma }{2} \right) + \frac{\gamma ^2}{2d_\gamma ^2}$$ 2 - γ d γ 2 γ + γ 2 + γ 2 2 d γ 2 (resp. $$d_\gamma -1$$ d γ - 1 ), where $$d_\gamma $$ d γ is the Hausdorff dimension of the whole plane with respect to the $$\gamma $$ γ -LQG metric. For $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , in which case $$d_{\sqrt{8/3}}=4$$ d 8 / 3 = 4 , we get that the essential supremum of Euclidean (resp. $$\sqrt{8/3}$$ 8 / 3 -LQG) dimension of a $$\sqrt{8/3}$$ 8 / 3 -LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and $$\gamma $$ γ -LQG Hausdorff dimensions of the intersection of a $$\gamma $$ γ -LQG ball boundary with the set of metric $$\alpha $$ α -thick points of the field h for each $$\alpha \in \mathbb R$$ α ∈ R . Our results show that the set of $$\gamma /d_\gamma $$ γ / d γ -thick points on the ball boundary has full Euclidean dimension and the set of $$\gamma $$ γ -thick points on the ball boundary has full $$\gamma $$ γ -LQG dimension.

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

2019 ◽  
Vol 176 (1-2) ◽  
pp. 649-667
Author(s):  
Ewain Gwynne ◽  
Nina Holden ◽  
Jason Miller
Keyword(s):  
Hausdorff Dimension ◽  
Free Field ◽  
Borel Subset ◽  
Intermediate Step ◽  
Conformal Map ◽  
Gaussian Free Field ◽  
Conformal Maps ◽  
Upper Half Plane ◽  
The One ◽  
The Relationship

Abstract We prove a formula relating the Hausdorff dimension of a deterministic Borel subset of $${\mathbb {R}}$$R and the Hausdorff dimension of its image under a conformal map from the upper half-plane to a complementary connected component of an $$\hbox {SLE}_\kappa $$SLEκ curve for $$\kappa \not =4$$κ≠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 $$\hbox {SLE}_\kappa $$SLEκ curve for $$\kappa \in (0,4)\cup (4,8)$$κ∈(0,4)∪(4,8) and the dimension of the same set with respect to the $$\gamma $$γ-quantum natural parameterization of the curve induced by an independent Gaussian free field, $$\gamma = \sqrt{\kappa }\wedge (4/\sqrt{\kappa })$$γ=κ∧(4/κ).

scholarly journals Crystallization of Random Matrix Orbits

2018 ◽  
Vol 2020 (3) ◽  
pp. 883-913 ◽  
Author(s):  
Vadim Gorin ◽  
Adam W Marcus
Keyword(s):  
Random Matrix ◽  
Special Functions ◽  
Free Field ◽  
Gaussian Free Field ◽  
Complex Quaternion ◽  
Cutting Out ◽  
Multiplicative Convolution ◽  
Global Asymptotics ◽  
Associated Special Functions ◽  
Derivatives Of

Abstract Three operations on eigenvalues of real/complex/quaternion (corresponding to $\beta =1,2,4$) matrices, obtained from cutting out principal corners, adding, and multiplying matrices, can be extrapolated to general values of $\beta>0$ through associated special functions. We show that the $\beta \to \infty $ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general $\beta $ self-adjoint matrix with fixed eigenvalues is known as the $\beta $-corners process. We show that as $\beta \to \infty $ these eigenvalues crystallize on an irregular lattice consisting of the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field put on top of this lattice, which provides a new explanation as to why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles.

scholarly journals The local structure of the free boundary in the fractional obstacle problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Matteo Focardi ◽  
Emanuele Spadaro
Keyword(s):  
Hausdorff Dimension ◽  
Free Boundary ◽  
Hausdorff Measure ◽  
Local Structure ◽  
Obstacle Problem ◽  
List Type ◽  
Minkowski Content ◽  
Local Finiteness ◽  
Lower Dimensional

AbstractBuilding upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in {\mathbb{R}^{n+1}} with obstacle function φ (suitably smooth and decaying fast at infinity) up to sets of null {{\mathcal{H}}^{n-1}} measure. In particular, if φ is analytic, the problem reduces to the zero obstacle case dealt with in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] and therefore we retrieve the same results:(i)local finiteness of the {(n-1)}-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure),(ii){{\mathcal{H}}^{n-1}}-rectifiability of the free boundary,(iii)classification of the frequencies and of the blowups up to a set of Hausdorff dimension at most {(n-2)} in the free boundary.Instead, if {\varphi\in C^{k+1}(\mathbb{R}^{n})}, {k\geq 2}, similar results hold only for distinguished subsets of points in the free boundary where the order of contact of the solution with the obstacle function φ is less than {k+1}.

scholarly journals Thick points of the Gaussian free field

2010 ◽  
Vol 38 (2) ◽  
pp. 896-926 ◽  
Author(s):  
Xiaoyu Hu ◽  
Jason Miller ◽  
Yuval Peres
Keyword(s):  
Free Field ◽  
Gaussian Free Field ◽  
Thick Points

scholarly journals Trends in deaths from road injuries during the COVID-19 pandemic in Japan, January to September 2020

2020 ◽  
Vol 7 (1) ◽  
Author(s):  
Shuhei Nomura ◽  
Takayuki Kawashima ◽  
Daisuke Yoneoka ◽  
Yuta Tanoue ◽  
Akifumi Eguchi ◽  
...  
Keyword(s):  
Lower Bound ◽  
Road Traffic ◽  
Prediction Interval ◽  
Road Transport ◽  
The United States ◽  
Point Estimate ◽  
Traffic Fatalities ◽  
State Of Emergency ◽  
Counter Measures ◽  
Mortality Burden

Abstract Background In Japan, the latest estimates of excess all-cause deaths through January to July 2020 showed that the overall (direct and indirect) mortality burden from the Coronavirus Disease 2019 (COVID-19) in Japan was relatively low compared to Europe and the United States. However, consistency between the reported number of COVID-19 deaths and excess all-cause deaths was limited across prefectures, suggesting the necessity of distinguishing the direct and indirect consequences of COVID-19 by cause-specific analysis. To examine whether deaths from road injuries decreased during the COVID-19 pandemic in Japan, consistent with a possible reduction of road transport activity connected to Japan’s state of emergency declaration, we estimated the exiguous deaths from road injuries in each week from January to September 2020 by 47 prefectures. Methods To estimate the expected weekly number of deaths from road injuries, a quasi-Poisson regression was applied to daily traffic fatalities data obtained from Traffic Accident Research and Data Analysis, Japan. We set two thresholds, point estimate and lower bound of the two-sided 95% prediction interval, for exiguous deaths, and report the range of differences between the observed number of deaths and each of these thresholds as exiguous deaths. Results Since January 2020, in a few weeks the observed deaths from road injuries fell below the 95% lower bound, such as April 6–12 (exiguous deaths 5–21, percent deficit 2.82–38.14), May 4–10 (8–23, 21.05–43.01), July 20–26 (12–29, 30.77–51.53), and August 3–9 (3–20, 7.32–34.41). However, those less than the 95% lower bound were also observed in weeks in the previous years. Conclusions The number of road traffic fatalities during the COVID-19 pandemic in Japan has decreased slightly, but not significantly, in several weeks compared with the average year. This suggests that the relatively small changes in excess all-cause mortality observed in Japan during the COVID-19 pandemic could not be explained simply by an offsetting reduction in traffic deaths. Considering a variety of other indirect effects, evaluating an independent, unbiased measure of COVID-19-related mortality burden could provide insight into the design of future broad-based infectious disease counter-measures and offer lessons to other countries.

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