We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble
We perform path-integral molecular dynamics (PIMD), ring-polymer MD (RPMD), and classical MD simulations of H
- Award ID(s):
- 2112550
- NSF-PAR ID:
- 10364848
- Publisher / Repository:
- Nature Publishing Group
- Date Published:
- Journal Name:
- Scientific Reports
- Volume:
- 12
- Issue:
- 1
- ISSN:
- 2045-2322
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract for$$\hbox {CLE}_{\kappa '}$$ in (4, 8) that is drawn on an independent$$\kappa '$$ -LQG surface for$$\gamma $$ . The results are similar in flavor to the ones from our companion paper dealing with$$\gamma ^2=16/\kappa '$$ for$$\hbox {CLE}_{\kappa }$$ 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$$\kappa $$ 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 “$$\hbox {CLE}_{\kappa '}$$ CLE Percolations ” described the law of interfaces obtained when coloring the loops of a independently into two colors with respective probabilities$$\hbox {CLE}_{\kappa '}$$ p and . This description was complete up to one missing parameter$$1-p$$ . The results of the present paper about CLE on LQG allow us to determine its value in terms of$$\rho $$ p and . It shows in particular that$$\kappa '$$ and$$\hbox {CLE}_{\kappa '}$$ are related via a continuum analog of the Edwards-Sokal coupling between$$\hbox {CLE}_{16/\kappa '}$$ percolation and the$$\hbox {FK}_q$$ q -state Potts model (which makes sense even for non-integerq between 1 and 4) if and only if . This provides further evidence for the long-standing belief that$$q=4\cos ^2(4\pi / \kappa ')$$ and$$\hbox {CLE}_{\kappa '}$$ represent the scaling limits of$$\hbox {CLE}_{16/\kappa '}$$ percolation and the$$\hbox {FK}_q$$ q -Potts model whenq and are related in this way. Another consequence of the formula for$$\kappa '$$ 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.$$\rho (p,\kappa ')$$ -
Abstract We provide moment bounds for expressions of the type
where$$(X^{(1)} \otimes \cdots \otimes X^{(d)})^T A (X^{(1)} \otimes \cdots \otimes X^{(d)})$$ denotes the Kronecker product and$$\otimes $$ are random vectors with independent, mean 0, variance 1, subgaussian entries. The bounds are tight up to constants depending on$$X^{(1)}, \ldots , X^{(d)}$$ d for the case of Gaussian random vectors. Our proof also provides a decoupling inequality for expressions of this type. Using these bounds, we obtain new, improved concentration inequalities for expressions of the form .$$\Vert B (X^{(1)} \otimes \cdots \otimes X^{(d)})\Vert _2$$ -
Abstract We present the first unquenched lattice-QCD calculation of the form factors for the decay
at nonzero recoil. Our analysis includes 15 MILC ensembles with$$B\rightarrow D^*\ell \nu $$ flavors of asqtad sea quarks, with a strange quark mass close to its physical mass. The lattice spacings range from$$N_f=2+1$$ fm down to 0.045 fm, while the ratio between the light- and the strange-quark masses ranges from 0.05 to 0.4. The valence$$a\approx 0.15$$ b andc quarks are treated using the Wilson-clover action with the Fermilab interpretation, whereas the light sector employs asqtad staggered fermions. We extrapolate our results to the physical point in the continuum limit using rooted staggered heavy-light meson chiral perturbation theory. Then we apply a model-independent parametrization to extend the form factors to the full kinematic range. With this parametrization we perform a joint lattice-QCD/experiment fit using several experimental datasets to determine the CKM matrix element . We obtain$$|V_{cb}|$$ . The first error is theoretical, the second comes from experiment and the last one includes electromagnetic and electroweak uncertainties, with an overall$$\left| V_{cb}\right| = (38.40 \pm 0.68_{\text {th}} \pm 0.34_{\text {exp}} \pm 0.18_{\text {EM}})\times 10^{-3}$$ , which illustrates the tensions between the experimental data sets, and between theory and experiment. This result is in agreement with previous exclusive determinations, but the tension with the inclusive determination remains. Finally, we integrate the differential decay rate obtained solely from lattice data to predict$$\chi ^2\text {/dof} = 126/84$$ , which confirms the current tension between theory and experiment.$$R(D^*) = 0.265 \pm 0.013$$ -
Abstract Using extensive numerical simulation of the Navier–Stokes equations, we study the transition from the Darcy’s law for slow flow of fluids through a disordered porous medium to the nonlinear flow regime in which the effect of inertia cannot be neglected. The porous medium is represented by two-dimensional slices of a three-dimensional image of a sandstone. We study the problem over wide ranges of porosity and the Reynolds number, as well as two types of boundary conditions, and compute essential features of fluid flow, namely, the strength of the vorticity, the effective permeability of the pore space, the frictional drag, and the relationship between the macroscopic pressure gradient
and the fluid velocity$${\varvec{\nabla }}P$$ v . The results indicate that when the Reynolds number Re is low enough that the Darcy’s law holds, the magnitude of the vorticity is nearly zero. As Re increases, however, so also does$$\omega _z$$ , and its rise from nearly zero begins at the same Re at which the Darcy’s law breaks down. We also show that a nonlinear relation between the macroscopic pressure gradient and the fluid velocity$$\omega _z$$ v , given by, , provides accurate representation of the numerical data, where$$-{\varvec{\nabla }}P=(\mu /K_e)\textbf{v}+\beta _n\rho |\textbf{v}|^2\textbf{v}$$ and$$\mu$$ are the fluid’s viscosity and density,$$\rho$$ is the effective Darcy permeability in the linear regime, and$$K_e$$ is a generalized nonlinear resistance. Theoretical justification for the relation is presented, and its predictions are also compared with those of the Forchheimer’s equation.$$\beta _n$$ -
Abstract We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in
and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$ , by showing that$\mathcal { C}$ admits non-trivial satisfiability and/or$\mathcal { C}$ # SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial# SAT algorithm for a circuit class . Say that a symmetric Boolean function${\mathcal C}$ f (x 1,…,x n ) issparse if it outputs 1 onO (1) values of . We show that for every sparse${\sum }_{i} x_{i}$ f , and for all “typical” , faster$\mathcal { C}$ # SAT algorithms for circuits imply lower bounds against the circuit class$\mathcal { C}$ , which may be$f \circ \mathcal { C}$ stronger than itself. In particular:$\mathcal { C}$ # SAT algorithms forn k -size -circuits running in 2$\mathcal { C}$ n /n k time (for allk ) implyN E X P does not have -circuits of polynomial size.$(f \circ \mathcal { C})$ # SAT algorithms for -size$2^{n^{{\varepsilon }}}$ -circuits running in$\mathcal { C}$ time (for some$2^{n-n^{{\varepsilon }}}$ ε > 0) implyQ u a s i -N P does not have -circuits of polynomial size.$(f \circ \mathcal { C})$ Applying
# SAT algorithms from the literature, one immediate corollary of our results is thatQ u a s i -N P does not haveE M A J ∘A C C 0∘T H R circuits of polynomial size, whereE M A J is the “exact majority” function, improving previous lower bounds againstA C C 0[Williams JACM’14] andA C C 0∘T H R [Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.