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1. Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

   https://doi.org/10.1007/s00440-021-01070-4  

   Miller, Jason ; Sheffield, Scott ; Werner, Wendelin ( June 2021 , Probability Theory and Related Fields)

   We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$for$$\kappa '$$${\kappa }^{\prime }$in (4, 8) that is drawn on an independent$$\gamma $$$\gamma$-LQG surface for$$\gamma ^2=16/\kappa '$$${\gamma }^2=16/{\kappa }^{\prime }$. The results are similar in flavor to the ones from our companion paper dealing with$$\hbox {CLE}_{\kappa }$$${\text{CLE}}_{\kappa }$for$$\kappa $$$\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$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$in terms of stable growth-fragmentation 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 “CLE Percolations” described the law of interfaces obtained when coloring the loops of a$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$independently into two colors with respective probabilitiespand$$1-p$$$1-p$. This description was complete up to one missing parameter$$\rho $$$\rho$. The results of the present paper about CLE on LQG allow us to determine its value in terms ofpand$$\kappa '$$${\kappa }^{\prime }$. It shows in particular that$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$and$$\hbox {CLE}_{16/\kappa '}$$${\text{CLE}}_{16/{\kappa }^{\prime }}$are related via a continuum analog of the Edwards-Sokal coupling between$$\hbox {FK}_q$$${\text{FK}}_q$percolation and theq-state Potts model (which makes sense evenmore »for non-integerqbetween 1 and 4) if and only if$$q=4\cos ^2(4\pi / \kappa ')$$$q=4{cos}^2(4\pi /{\kappa }^{\prime })$. This provides further evidence for the long-standing belief that$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$and$$\hbox {CLE}_{16/\kappa '}$$${\text{CLE}}_{16/{\kappa }^{\prime }}$represent the scaling limits of$$\hbox {FK}_q$$${\text{FK}}_q$percolation and theq-Potts model whenqand$$\kappa '$$${\kappa }^{\prime }$are related in this way. Another consequence of the formula for$$\rho (p,\kappa ')$$$\rho (p,{\kappa }^{\prime })$is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

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2. Green function estimates on complements of low-dimensional uniformly rectifiable sets

   https://doi.org/10.1007/s00208-022-02379-8  

   David, Guy ; Feneuil, Joseph ; Mayboroda, Svitlana ( March 2022 , Mathematische Annalen)

   It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$$L_{\beta ,\gamma }=-\text{div}{D}^{d+1+\gamma -n}\nabla$associated to a domain$$\Omega \subset {\mathbb {R}}^n$$$\Omega \subset R^n$with a uniformly rectifiable boundary$$\Gamma $$$\Gamma$of dimension$$d < n-1$$$d<n-1$, the now usual distance to the boundary$$D = D_\beta $$$D=D_{\beta }$given by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$$D_{\beta }{\left(X\right)}^{-\beta }={\int }_{\Gamma }{|X-y|}^{-d-\beta }d\sigma \left(y\right)$for$$X \in \Omega $$$X\in \Omega$, where$$\beta >0$$$\beta >0$and$$\gamma \in (-1,1)$$$\gamma \in (-1,1)$. In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$$L_{\beta ,\gamma }$, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$$D^{1-\gamma }$, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$$|D\nabla (ln(\frac{G}{D^{1-\gamma }})){|}^2$satisfies a Carleson measure estimate on$$\Omega $$$\Omega$. We underline that the strong and the weak results are different in nature and, of course, at the levelmore »of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).

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3. Softmax policy gradient methods can take exponential time to converge

   https://doi.org/10.1007/s10107-022-01920-6  

   Li, Gen ; Wei, Yuting ; Chi, Yuejie ; Chen, Yuxin ( January 2023 , Mathematical Programming)

   The softmax policy gradient (PG) method, which performs gradient ascent under softmax policy parameterization, is arguably one of the de facto implementations of policy optimization in modern reinforcement learning. For$$\gamma $$$\gamma$-discounted infinite-horizon tabular Markov decision processes (MDPs), remarkable progress has recently been achieved towards establishing global convergence of softmax PG methods in finding a near-optimal policy. However, prior results fall short of delineating clear dependencies of convergence rates on salient parameters such as the cardinality of the state space$${\mathcal {S}}$$$S$and the effective horizon$$\frac{1}{1-\gamma }$$$\frac{1}{1-\gamma }$, both of which could be excessively large. In this paper, we deliver a pessimistic message regarding the iteration complexity of softmax PG methods, despite assuming access to exact gradient computation. Specifically, we demonstrate that the softmax PG method with stepsize$$\eta $$$\eta$can take$$\begin{aligned} \frac{1}{\eta } |{\mathcal {S}}|^{2^{\Omega \big (\frac{1}{1-\gamma }\big )}} ~\text {iterations} \end{aligned}$$$\begin{array}{c}\frac{1}{\eta }{\left|S\right|}^{2^{\Omega (\frac{1}{1-\gamma })}}\text{iterations}\end{array}$to converge, even in the presence of a benign policy initialization and an initial state distribution amenable to exploration (so that the distribution mismatch coefficient is not exceedingly large). This is accomplished by characterizing the algorithmic dynamics over a carefully-constructed MDP containing only three actions. Our exponential lower bound hints at the necessity of carefully adjusting update rules or enforcing proper regularization inmore »accelerating PG methods.

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4. Plasmonically enhanced mid-IR light source based on tunable spectrally and directionally selective thermal emission from nanopatterned graphene

   https://doi.org/10.1038/s41598-020-73582-3  

   Shabbir, Muhammad Waqas ; Leuenberger, Michael N. ( October 2020 , Scientific Reports)

   We present a proof of concept for a spectrally selective thermal mid-IR source based on nanopatterned graphene (NPG) with a typical mobility of CVD-grown graphene (up to 3000$$\hbox {cm}^2\,\hbox {V}^{-1}\,\hbox {s}^{-1}$$${\text{cm}}^2{\text{V}}^{-1}{\text{s}}^{-1}$), ensuring scalability to large areas. For that, we solve the electrostatic problem of a conducting hyperboloid with an elliptical wormhole in the presence of anin-planeelectric 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$$\lambda =3$$$\lambda =3$to 12$$\upmu$$$\mu$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 state-of-the-art pristine graphene light sources operating in the near-infrared by at least a factor of 100. According to our COMSOL calculations, a maximum emission power per area of$$11\times 10^3$$$11\times {10}^3$W/$$\hbox {m}^2$$${\text{m}}^2$at$$T=2000$$$T=2000$K for a bias voltage of$$V=23$$$V=23$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 derivingmore »the completely general nonlocal fluctuation-dissipation 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$$\upmu$$$\mu$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$$12^\circ$$${12}^{\circ }$and$$80^\circ$$${80}^{\circ }$by tuning the Fermi energy between$$E_F=1.0$$$E_F=1.0$eV and$$E_F=0.25$$$E_F=0.25$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 frequency-dependent. Using finite-difference time domain calculations, coupled mode theory, and RPA, we develop the model of a mid-IR light source based on NPG, which will pave the way to graphene-based optical mid-IR communication, mid-IR color displays, mid-IR spectroscopy, and virus detection.

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5. Neutrino emission from binary neutron star mergers: characterising light curves and mean energies

   https://doi.org/10.1140/epja/s10050-022-00743-5  

   Cusinato, Marco ; Guercilena, Federico Maria ; Perego, Albino ; Logoteta, Domenico ; Radice, David ; Bernuzzi, Sebastiano ; Ansoldi, Stefano ( May 2022 , The European Physical Journal A)

   Neutrinos are copiously emitted by neutron star mergers, due to the high temperatures reached by dense matter during the merger and its aftermath. Neutrinos influence the merger dynamics and shape the properties of the ejecta, including the resultingr-process nucleosynthesis and kilonova emission. In this work, we analyse neutrino emission from a large sample of binary neutron star merger simulations in Numerical Relativity, covering a broad range of initial masses, nuclear equation of state and viscosity treatments. We extract neutrino luminosities and mean energies, and compute quantities of interest such as the peak values, peak broadnesses, time averages and decrease time scales. We provide a systematic description of such quantities, including their dependence on the initial parameters of the system. We find that for equal-mass systems the total neutrino luminosity (several$$10^{53}{\hbox {erg}~{\hbox {s}}^{-1}}$$${10}^{53}\text{erg}{\text{s}}^{-1}$) decreases as the reduced tidal deformability increases, as a consequence of the less violent merger dynamics. Similarly, tidal disruption in asymmetric mergers leads to systematically smaller luminosities. Peak luminosities can be twice as large as the average ones. Electron antineutrino luminosities dominate (initially by a factor of 2-3) over electron neutrino ones, while electron neutrinos and heavy flavour neutrinos have similar luminosities. Mean energies are nearly constantmore »in time and independent on the binary parameters. Their values reflect the different decoupling temperature inside the merger remnant. Despite present uncertainties in neutrino modelling, our results provide a broad and physically grounded characterisation of neutrino emission, and they can serve as a reference point to develop more sophisticated neutrino transport schemes.

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