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1. Weyl remainders: an application of geodesic beams

   https://doi.org/10.1007/s00222-023-01178-5  

   Canzani, Yaiza; Galkowski, Jeffrey (June 2023, Inventiones mathematicae)

   Abstract We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold ( M ,  g ) of dimension n , let $$\Pi _\lambda $$ Π λ denote the kernel of the spectral projector for the Laplacian, $$\mathbb {1}_{[0,\lambda ^2]}(-\Delta _g)$$ 1 [ 0 , λ 2 ] ( - Δ g ) . Assuming only that the set of near periodic geodesics over $${W}\subset M$$ W ⊂ M has small measure, we prove that as $$\lambda \rightarrow \infty $$ λ → ∞ $$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned}$$ ∫ W Π λ ( x , x ) d x = ( 2 π ) - n vol R n ( B ) vol g ( W ) λ n + O ( λ n - 1 log λ ) , where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $$\Pi _\lambda (x,y)$$ Π λ ( x , y ) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams. 

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2. Critical value asymptotics for the contact process on random graphs

   https://doi.org/10.1090/tran/8399  

   Nam, Danny; Nguyen, Oanh; Sly, Allan (January 2022, Transactions of the American Mathematical Society)

   Recent progress in the study of the contact process (see Shankar Bhamidi, Danny Nam, Oanh Nguyen, and Allan Sly [Ann. Probab. 49 (2021), pp. 244–286]) has verified that the extinction-survival threshold λ 1 \lambda _1 on a Galton-Watson tree is strictly positive if and only if the offspring distribution ξ \xi has an exponential tail. In this paper, we derive the first-order asymptotics of λ 1 \lambda _1 for the contact process on Galton-Watson trees and its corresponding analog for random graphs. In particular, if ξ \xi is appropriately concentrated around its mean, we demonstrate that λ 1 ( ξ ) ∼ 1 / E ξ \lambda _1(\xi ) \sim 1/\mathbb {E} \xi as E ξ → ∞ \mathbb {E}\xi \rightarrow \infty , which matches with the known asymptotics on d d -regular trees. The same results for the short-long survival threshold on the Erdős-Rényi and other random graphs are shown as well. 

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3. Scattering interference signature of a pair density wave state in the cuprate pseudogap phase

   https://doi.org/10.1038/s41467-021-26028-x  

   Wang, Shuqiu; Choubey, Peayush; Chong, Yi Xue; Chen, Weijiong; Ren, Wangping; Eisaki, H.; Uchida, S.; Hirschfeld, Peter J.; Davis, J. C. (December 2021, Nature Communications)

   Abstract An unidentified quantum fluid designated the pseudogap (PG) phase is produced by electron-density depletion in the CuO 2 antiferromagnetic insulator. Current theories suggest that the PG phase may be a pair density wave (PDW) state characterized by a spatially modulating density of electron pairs. Such a state should exhibit a periodically modulating energy gap $${\Delta }_{{{{{{\rm{P}}}}}}}({{{{{\boldsymbol{r}}}}}})$$ Δ P ( r ) in real-space, and a characteristic quasiparticle scattering interference (QPI) signature $${\Lambda }_{{{{{{\rm{P}}}}}}}({{{{{\boldsymbol{q}}}}}})$$ Λ P ( q ) in wavevector space. By studying strongly underdoped Bi 2 Sr 2 CaDyCu 2 O 8 at hole-density ~0.08 in the superconductive phase, we detect the 8 a 0 -periodic $${\Delta }_{{{{{{\rm{P}}}}}}}({{{{{\boldsymbol{r}}}}}})$$ Δ P ( r ) modulations signifying a PDW coexisting with superconductivity. Then, by visualizing the temperature dependence of this electronic structure from the superconducting into the pseudogap phase, we find the evolution of the scattering interference signature $$\Lambda ({{{{{\boldsymbol{q}}}}}})$$ Λ ( q ) that is predicted specifically for the temperature dependence of an 8 a 0 -periodic PDW. These observations are consistent with theory for the transition from a PDW state coexisting with d -wave superconductivity to a pure PDW state in the Bi 2 Sr 2 CaDyCu 2 O 8 pseudogap phase. 

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4. Anisotropic flow and flow fluctuations of identified hadrons in Pb–Pb collisions at $$ \sqrt{s_{\textrm{NN}}} $$ = 5.02 TeV

   https://doi.org/10.1007/JHEP05(2023)243  

   Acharya, S.; Adamová, D.; Adler, A.; Aglieri Rinella, G.; Agnello, M.; Agrawal, N.; Ahammed, Z.; Ahmad, S.; Ahn, S. U.; Ahuja, I.; et al (June 2023, Journal of High Energy Physics)

   A bstract The first measurements of elliptic flow of π ± , K ± , $$ \textrm{p}+\overline{\textrm{p}} $$ p + p ¯ , $$ {\textrm{K}}_{\textrm{S}}^0 $$ K S 0 , $$ \Lambda +\overline{\Lambda} $$ Λ + Λ ¯ , ϕ , $$ {\Xi}^{-}+{\overline{\Xi}}^{+} $$ Ξ − + Ξ ¯ + , and $$ {\varOmega}^{-}+{\overline{\varOmega}}^{+} $$ Ω − + Ω ¯ + using multiparticle cumulants in Pb–Pb collisions at $$ \sqrt{s_{\textrm{NN}}} $$ s NN = 5 . 02 TeV are resented. Results obtained with two- ( v 2 {2}) and four-particle cumulants ( v 2 {4}) are shown as a function of transverse momentum, p T , for various collision centrality intervals. Combining the data for both v 2 {2} and v 2 {4} also allows us to report the first measurements of the mean elliptic flow, elliptic flow fluctuations, and relative elliptic flow fluctuations for various hadron species. These observables probe the event-by-event eccentricity fluctuations in the initial state and the contributions from the dynamic evolution of the expanding quark–gluon plasma. The characteristic features observed in previous p T -differential anisotropic flow measurements for identified hadrons with two-particle correlations, namely the mass ordering at low p T and the approximate scaling with the number of constituent quarks at intermediate p T , are similarly present in the four-particle correlations and the combinations of v 2 {2} and v 2 {4}. In addition, a particle species dependence of flow fluctuations is observed that could indicate a significant contribution from final state hadronic interactions. The comparison between experimental measurements and CoLBT model calculations, which combine the various physics processes of hydrodynamics, quark coalescence, and jet fragmentation, illustrates their importance over a wide p T range. 

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5. Observation of $$ {\Lambda}_b^0 $$ → D+pπ−π− and $$ {\Lambda}_b^0 $$ → D*+pπ−π− decays

   https://doi.org/10.1007/JHEP03(2022)153  

   Aaij, R.; Abdelmotteleb, A. S.; Abellán Beteta, C.; Abudinén, F.; Ackernley, T.; Adeva, B.; Adinolfi, M.; Afsharnia, H.; Agapopoulou, C.; Aidala, C. A.; et al (March 2022, Journal of High Energy Physics)

   Abstract The multihadron decays $$ {\Lambda}_b^0 $$ Λ b 0 → D + pπ−π− and $$ {\Lambda}_b^0 $$ Λ b 0 → D * + pπ−π− are observed in data corresponding to an integrated luminosity of 3 fb − 1 , collected in proton-proton collisions at centre-of-mass energies of 7 and 8 TeV by the LHCb detector. Using the decay $$ {\Lambda}_b^0 $$ Λ b 0 → $$ {\Lambda}_c^{+} $$ Λ c + π + π − π − as a normalisation channel, the ratio of branching fractions is measured to be $$ \frac{\mathcal{B}\left({\Lambda}_b^0\to {D}^{+}p{\pi}^{-}{\pi}^{-}\right)}{\mathcal{B}\left({\Lambda}_b^0\to {\Lambda}_c^0{\pi}^{+}{\pi}^{-}{\pi}^{-}\right)}\times \frac{\mathcal{B}\left({D}^{+}\to {K}^{-}{\pi}^{+}{\pi}^{+}\right)}{\mathcal{B}\left({\Lambda}_c^0\to {pK}^{-}{\pi}^{-}\right)}=\left(5.35\pm 0.21\pm 0.16\right)\%, $$ B Λ b 0 → D + p π − π − B Λ b 0 → Λ c 0 π + π − π − × B D + → K − π + π + B Λ c 0 → pK − π − = 5.35 ± 0.21 ± 0.16 % , where the first uncertainty is statistical and the second systematic. The ratio of branching fractions for the $$ {\Lambda}_b^0 $$ Λ b 0 → D *+ pπ − π − and $$ {\Lambda}_b^0 $$ Λ b 0 → D + pπ − π − decays is found to be $$ \frac{\mathcal{B}\left({\Lambda}_b^0\to {D}^{\ast +}p{\pi}^{-}{\pi}^{-}\right)}{\mathcal{B}\left({\Lambda}_b^0\to {D}^{+}p{\pi}^{-}{\pi}^{-}\right)}\times \left(\mathcal{B}\left({D}^{\ast +}\to {D}^{+}{\pi}^0\right)+\mathcal{B}\left({D}^{\ast +}\to {D}^{+}\gamma \right)\right)=\left(61.3\pm 4.3\pm 4.0\right)\%. $$ B Λ b 0 → D ∗ + p π − π − B Λ b 0 → D + p π − π − × B D ∗ + → D + π 0 + B D ∗ + → D + γ = 61.3 ± 4.3 ± 4.0 % . 

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