skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Weak LQG metrics and Liouville first passage percolation
Abstract For $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , we define a weak $$\gamma $$ γ - Liouville quantum gravity ( LQG ) metric to be a function $$h\mapsto D_h$$ h ↦ D h which takes in an instance of the planar Gaussian free field and outputs a metric on the plane satisfying a certain list of natural axioms. We show that these axioms are satisfied for any subsequential limits of Liouville first passage percolation. Such subsequential limits were proven to exist by Ding et al. (Tightness of Liouville first passage percolation for $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , 2019. ArXiv e-prints, arXiv:1904.08021 ). It is also known that these axioms are satisfied for the $$\sqrt{8/3}$$ 8 / 3 -LQG metric constructed by Miller and Sheffield (2013–2016). For any weak $$\gamma $$ γ -LQG metric, we obtain moment bounds for diameters of sets as well as point-to-point, set-to-set, and point-to-set distances. We also show that any such metric is locally bi-Hölder continuous with respect to the Euclidean metric and compute the optimal Hölder exponents in both directions. Finally, we show that LQG geodesics cannot spend a long time near a straight line or the boundary of a metric ball. These results are used in subsequent work by Gwynne and Miller which proves that the weak $$\gamma $$ γ -LQG metric is unique for each $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , which in turn gives the uniqueness of the subsequential limit of Liouville first passage percolation. However, most of our results are new even in the special case when $$\gamma =\sqrt{8/3}$$ γ = 8 / 3 .  more » « less
Award ID(s):
1811092
PAR ID:
10226346
Author(s) / Creator(s):
; ; ; ;
Date Published:
Journal Name:
Probability Theory and Related Fields
Volume:
178
Issue:
1-2
ISSN:
0178-8051
Page Range / eLocation ID:
369 to 436
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; (, Advanced Nonlinear Studies)
    Abstract We investigate the rigidity of global minimizers u ≥ 0 u\ge 0 of the Alt-Phillips functional involving negative power potentials ∫ Ω ( ∣ ∇ u ∣ 2 + u − γ χ { u > 0 } ) d x , γ ∈ ( 0 , 2 ) , \mathop{\int }\limits_{\Omega }(| \nabla u{| }^{2}+{u}^{-\gamma }{\chi }_{\left\{u\gt 0\right\}}){\rm{d}}x,\hspace{1.0em}\gamma \in \left(0,2), when the exponent γ \gamma is close to the extremes of the admissible values. In particular, we show that global minimizers in R n {{\mathbb{R}}}^{n} are one-dimensional if γ \gamma is close to 2 and n ≤ 7 n\le 7 , or if γ \gamma is close to 0 and n ≤ 4 n\le 4 . 
    more » « less
  2. ; (, Journal of High Energy Physics)
    A<sc>bstract</sc> We study the “three particle coupling”$$ {\Gamma}_{11}^1\left(\xi \right) $$ Γ 11 1 ξ , in 2dIsing Field Theory in a magnetic field, as the function of the scaling parameterξ:=h/(−m)15/8, wherem∼Tc−Tandh∼Hare scaled deviation from the critical temperature and scaled external field, respectively. The “φ3coupling”$$ {\Gamma}_{11}^1 $$ Γ 11 1 is defined in terms of the residue of the 2 → 2 elastic scattering amplitude at its pole associated with the lightest particle itself. We limit attention to the High-Temperature domain, so thatmis negative. We suggest “standard analyticity”:$$ {\left({\Gamma}_{11}^1\right)}^2 $$ Γ 11 1 2 , as the function ofu:=ξ2, is analytic in the whole complexu-plane except for the branch cut from – ∞ to –u0≈ – 0.03585, the latter branching point –u0being associated with the Yang-Lee edge singularity. Under this assumption, the values of$$ {\Gamma}_{11}^1 $$ Γ 11 1 at any complexuare expressed through the discontinuity across the branch cut. We suggest approximation for this discontinuity which accounts for singular expansion of$$ {\Gamma}_{11}^1 $$ Γ 11 1 near the Yang-Lee branching point, as well as its known asymptotic atu →+∞. The resulting dispersion relation agrees well with known exact data, and with numerics obtained via Truncated Free Fermion Space Approach. This work is part of extended project of studying the S-matrix of the Ising Field Theory in a magnetic field. 
    more » « less
  3. ; (, ArXivorg)
    We study first passage percolation (FPP) with stationary edge weights on Cayley graphs of finitely generated virtually nilpotent groups. Previous works of Benjamini-Tessera and Cantrell-Furman show that scaling limits of such FPP are given by Carnot-Carathéodory metrics on the associated graded nilpotent Lie group. We show a converse, i.e. that for any Cayley graph of a finitely generated nilpotent group, any Carnot-Carathéodory metric on the associated graded nilpotent Lie group is the scaling limit of some FPP with stationary edge weights on that graph. Moreover, for any Cayley graph of any finitely generated virtually nilpotent group, any conjugation-invariant metric is the scaling limit of some FPP with stationary edge weights on that graph. We also show that the conjugation-invariant condition is also a necessary condition in all cases where scaling limits are known to exist. 
    more » « less
  4. ; ; ; ; ; ; ; ; ; ; et al (, Journal of High Energy Physics)
    A<sc>bstract</sc> A search for the fully reconstructed$$ {B}_s^0 $$ B s 0 → μ+μ−γdecay is performed at the LHCb experiment using proton-proton collisions at$$ \sqrt{s} $$ s = 13 TeV corresponding to an integrated luminosity of 5.4 fb−1. No significant signal is found and upper limits on the branching fraction in intervals of the dimuon mass are set$$ {\displaystyle \begin{array}{cc}\mathcal{B}\left({B}_s^0\to {\mu}^{+}{\mu}^{-}\gamma \right)<4.2\times {10}^{-8},& m\left({\mu}^{+}{\mu}^{-}\right)\in \left[2{m}_{\mu },1.70\right]\textrm{GeV}/{c}^2,\\ {}\mathcal{B}\left({B}_s^0\to {\mu}^{+}{\mu}^{-}\gamma \right)<7.7\times {10}^{-8},&\ m\left({\mu}^{+}{\mu}^{-}\right)\in \left[\textrm{1.70,2.88}\right]\textrm{GeV}/{c}^2,\\ {}\mathcal{B}\left({B}_s^0\to {\mu}^{+}{\mu}^{-}\gamma \right)<4.2\times {10}^{-8},& m\left({\mu}^{+}{\mu}^{-}\right)\in \left[3.92,{m}_{B_s^0}\right]\textrm{GeV}/{c}^2,\end{array}} $$ B B s 0 → μ + μ − γ < 4.2 × 10 − 8 , m μ + μ − ∈ 2 m μ 1.70 GeV / c 2 , B B s 0 → μ + μ − γ < 7.7 × 10 − 8 , m μ + μ − ∈ 1.70, 2.88 GeV / c 2 , B B s 0 → μ + μ − γ < 4.2 × 10 − 8 , m μ + μ − ∈ 3.92 m B s 0 GeV / c 2 , at 95% confidence level. Additionally, upper limits are set on the branching fraction in the [2mμ,1.70] GeV/c2dimuon mass region excluding the contribution from the intermediateϕ(1020) meson, and in the region combining all dimuon-mass intervals. 
    more » « less
  5. (, SciPost Physics Proceedings)
    As the lightest meson, pion offers unique opportunities for measuring parameters and testing limits of the Standard Model (SM). The PiBeta experiment, carried out at PSI, focused on SM tests accessible through the pion beta, \pi^+ \to \pi^0e^+\nu_e(\gamma) π + → π 0 e + ν e ( γ ) , and electronic radiative, \pi^+ \to e^+\nu_e\gamma π + → e + ν e γ , decay channels. We review the PiBeta experiment, and update the pion beta decay branching ratio B^{\text{exp}}_{\pi\beta}=1.038(6)_{\text{tot}}\times10^{-8} B π β exp = 1.038 ( 6 ) tot × 10 − 8 , along with the corresponding derived value of the Cabibbo-Kobayashi-Maskawa matrix element V_{ud} = 0.9738(28) V u d = 0.9738 ( 28 ) . 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email