We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loopensemble
We prove that the Hilbert scheme of
 Award ID(s):
 2203823
 NSFPAR ID:
 10462915
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Communications in Mathematical Physics
 Volume:
 403
 Issue:
 2
 ISSN:
 00103616
 Page Range / eLocation ID:
 p. 10051068
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract for$$\hbox {CLE}_{\kappa '}$$ ${\text{CLE}}_{{\kappa}^{\prime}}$ in (4, 8) that is drawn on an independent$$\kappa '$$ ${\kappa}^{\prime}$ LQG surface for$$\gamma $$ $\gamma $ . The results are similar in flavor to the ones from our companion paper dealing with$$\gamma ^2=16/\kappa '$$ ${\gamma}^{2}=16/{\kappa}^{\prime}$ for$$\hbox {CLE}_{\kappa }$$ ${\text{CLE}}_{\kappa}$ in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the$$\kappa $$ $\kappa $ in terms of stable growthfragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: In our paper entitled “$$\hbox {CLE}_{\kappa '}$$ ${\text{CLE}}_{{\kappa}^{\prime}}$CLE Percolations ” described the law of interfaces obtained when coloring the loops of a independently into two colors with respective probabilities$$\hbox {CLE}_{\kappa '}$$ ${\text{CLE}}_{{\kappa}^{\prime}}$p and . This description was complete up to one missing parameter$$1p$$ $1p$ . The results of the present paper about CLE on LQG allow us to determine its value in terms of$$\rho $$ $\rho $p and . It shows in particular that$$\kappa '$$ ${\kappa}^{\prime}$ and$$\hbox {CLE}_{\kappa '}$$ ${\text{CLE}}_{{\kappa}^{\prime}}$ are related via a continuum analog of the EdwardsSokal coupling between$$\hbox {CLE}_{16/\kappa '}$$ ${\text{CLE}}_{16/{\kappa}^{\prime}}$ percolation and the$$\hbox {FK}_q$$ ${\text{FK}}_{q}$q state Potts model (which makes sense even for nonintegerq between 1 and 4) if and only if . This provides further evidence for the longstanding belief that$$q=4\cos ^2(4\pi / \kappa ')$$ $q=4{cos}^{2}(4\pi /{\kappa}^{\prime})$ and$$\hbox {CLE}_{\kappa '}$$ ${\text{CLE}}_{{\kappa}^{\prime}}$ represent the scaling limits of$$\hbox {CLE}_{16/\kappa '}$$ ${\text{CLE}}_{16/{\kappa}^{\prime}}$ percolation and the$$\hbox {FK}_q$$ ${\text{FK}}_{q}$q Potts model whenq and are related in this way. Another consequence of the formula for$$\kappa '$$ ${\kappa}^{\prime}$ is the value of halfplane arm exponents for such divideandcolor models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for twodimensional models.$$\rho (p,\kappa ')$$ $\rho (p,{\kappa}^{\prime})$ 
Abstract We present a proof of concept for a spectrally selective thermal midIR source based on nanopatterned graphene (NPG) with a typical mobility of CVDgrown graphene (up to 3000
), ensuring scalability to large areas. For that, we solve the electrostatic problem of a conducting hyperboloid with an elliptical wormhole in the presence of an$$\hbox {cm}^2\,\hbox {V}^{1}\,\hbox {s}^{1}$$ ${\text{cm}}^{2}\phantom{\rule{0ex}{0ex}}{\text{V}}^{1}\phantom{\rule{0ex}{0ex}}{\text{s}}^{1}$inplane electric field. The localized surface plasmons (LSPs) on the NPG sheet, partially hybridized with graphene phonons and surface phonons of the neighboring materials, allow for the control and tuning of the thermal emission spectrum in the wavelength regime from to 12$$\lambda =3$$ $\lambda =3$ m by adjusting the size of and distance between the circular holes in a hexagonal or square lattice structure. Most importantly, the LSPs along with an optical cavity increase the emittance of graphene from about 2.3% for pristine graphene to 80% for NPG, thereby outperforming stateoftheart pristine graphene light sources operating in the nearinfrared by at least a factor of 100. According to our COMSOL calculations, a maximum emission power per area of$$\upmu$$ $\mu $ W/$$11\times 10^3$$ $11\times {10}^{3}$ at$$\hbox {m}^2$$ ${\text{m}}^{2}$ K for a bias voltage of$$T=2000$$ $T=2000$ V is achieved by controlling the temperature of the hot electrons through the Joule heating. By generalizing Planck’s theory to any grey body and deriving the completely general nonlocal fluctuationdissipation theorem with nonlocal response of surface plasmons in the random phase approximation, we show that the coherence length of the graphene plasmons and the thermally emitted photons can be as large as 13$$V=23$$ $V=23$ m and 150$$\upmu$$ $\mu $ m, respectively, providing the opportunity to create phased arrays made of nanoantennas represented by the holes in NPG. The spatial phase variation of the coherence allows for beamsteering of the thermal emission in the range between$$\upmu$$ $\mu $ and$$12^\circ$$ ${12}^{\circ}$ by tuning the Fermi energy between$$80^\circ$$ ${80}^{\circ}$ eV and$$E_F=1.0$$ ${E}_{F}=1.0$ eV through the gate voltage. Our analysis of the nonlocal hydrodynamic response leads to the conjecture that the diffusion length and viscosity in graphene are frequencydependent. Using finitedifference time domain calculations, coupled mode theory, and RPA, we develop the model of a midIR light source based on NPG, which will pave the way to graphenebased optical midIR communication, midIR color displays, midIR spectroscopy, and virus detection.$$E_F=0.25$$ ${E}_{F}=0.25$ 
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