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1. Continuous phase transitions on Galton–Watson trees

   https://doi.org/10.1017/S0963548321000237  

   Johnson, Tobias ( March 2022 , Combinatorics, Probability and Computing)

   Abstract Distinguishing between continuous and first-order phase transitions is a major challenge in random discrete systems. We study the topic for events with recursive structure on Galton–Watson trees. For example, let $\mathcal{T}_1$ be the event that a Galton–Watson tree is infinite and let $\mathcal{T}_2$ be the event that it contains an infinite binary tree starting from its root. These events satisfy similar recursive properties: $\mathcal{T}_1$ holds if and only if $\mathcal{T}_1$ holds for at least one of the trees initiated by children of the root, and $\mathcal{T}_2$ holds if and only if $\mathcal{T}_2$ holds for at least two of these trees. The probability of $\mathcal{T}_1$ has a continuous phase transition, increasing from 0 when the mean of the child distribution increases above 1. On the other hand, the probability of $\mathcal{T}_2$ has a first-order phase transition, jumping discontinuously to a non-zero value at criticality. Given the recursive property satisfied by the event, we describe the critical child distributions where a continuous phase transition takes place. In many cases, we also characterise the event undergoing the phase transition. 

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2. Physical limits to sensing material properties

   https://doi.org/10.1038/s41467-020-18995-4  

   Beroz, Farzan ; Zhou, Di ; Mao, Xiaoming ; Lubensky, David K. ( December 2020 , Nature Communications)

   null (Ed.)

   Abstract All materials respond heterogeneously at small scales, which limits what a sensor can learn. Although previous studies have characterized measurement noise arising from thermal fluctuations, the limits imposed by structural heterogeneity have remained unclear. In this paper, we find that the least fractional uncertainty with which a sensor can determine a material constant λ 0 of an elastic medium is approximately $$\delta {\lambda }_{0}/{\lambda }_{0} \sim ({\Delta }_{\lambda }^{1/2}/{\lambda }_{0}){(d/a)}^{D/2}{(\xi /a)}^{D/2}$$ δ λ 0 / λ 0 ~ ( Δ λ 1 / 2 / λ 0 ) ( d / a ) D / 2 ( ξ / a ) D / 2 for a  ≫  d  ≫  ξ , $${\lambda }_{0}\gg {\Delta }_{\lambda }^{1/2}$$ λ 0 ≫ Δ λ 1 / 2 , and D  > 1, where a is the size of the sensor, d is its spatial resolution, ξ is the correlation length of fluctuations in λ 0 , Δ λ is the local variability of λ 0 , and D is the dimension of the medium. Our results reveal how one can construct devices capable of sensing near these limits, e.g. for medical diagnostics. We use our theoretical framework to estimate the limits of mechanosensing in a biopolymer network, a sensory process involved in cellular behavior, medical diagnostics, and material fabrication. 

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3. Marginal dynamics of interacting diffusions on unimodular Galton–Watson trees

   https://doi.org/10.1007/s00440-023-01226-4  

   Lacker, Daniel ; Ramanan, Kavita ; Wu, Ruoyu ( December 2023 , Probability Theory and Related Fields)

   Consider a system of homogeneous interacting diffusive particles labeled by the nodes of a unimodular Galton–Watson tree, where the state of each node evolves infinitesi- mally like a d-dimensional diffusion whose drift coefficient depends on (the histories of) its own state and the states of neighboring nodes, and whose diffusion coefficient depends only on (the history of) its own state. Under suitable regularity assumptions on the coefficients, an autonomous characterization is obtained for the marginal dis- tribution of the dynamics of the neighborhood of a typical node in terms of a certain local equation, which is a new kind of stochastic differential equation that is nonlinear in the sense of McKean. This equation describes a finite-dimensional non-Markovian stochastic process whose infinitesimal evolution at any time depends not only on the structure and current state of the neighborhood, but also on the conditional law of the current state given the past of the states of neighborhing nodes until that time. Such marginal distributions are of interest because they arise as weak limits of both marginal distributions and empirical measures of interacting diffusions on many sequences of sparse random graphs, including the configuration model and Erdös–Rényi graphs whose average degrees converge to a finite non-zero limit. The results obtained complement classical results in the mean-field regime, which characterize the limiting dynamics of homogeneous interacting diffusions on complete graphs, as the num- ber of nodes goes to infinity, in terms of a corresponding nonlinear Markov process. However, in the sparse graph setting, the topology of the graph strongly influences the dynamics, and the analysis requires a completely different approach. The proofs of existence and uniqueness of the local equation rely on delicate new conditional independence and symmetry properties of particle trajectories on unimodular Galton– Watson trees, as well as judicious use of changes of measure. 

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4. Measurements of branching fractions and asymmetry parameters of $$ {\Xi}_c^0\to \Lambda {\overline{K}}^{\ast 0} $$, $$ {\Xi}_c^0\to {\Sigma}^0{\overline{K}}^{\ast 0} $$, and $$ {\Xi}_c^0\to {\Sigma}^{+}{K}^{\ast -} $$ decays at Belle

   https://doi.org/10.1007/JHEP06(2021)160  

   Jia, S. ; Tang, S. S. ; Shen, C. P. ; Adachi, I. ; Aihara, H. ; Al Said, S. ; Asner, D. M. ; Aulchenko, V. ; Aushev, T. ; Ayad, R. ; et al ( June 2021 , Journal of High Energy Physics)

   A bstract Using a data sample of 980 fb − 1 collected with the Belle detector at the KEKB asymmetric-energy e + e − collider, we study the processes of $$ {\Xi}_c^0\to \Lambda {\overline{K}}^{\ast 0} $$ Ξ c 0 → Λ K ¯ ∗ 0 , $$ {\Xi}_c^0\to {\Sigma}^0{\overline{K}}^{\ast 0} $$ Ξ c 0 → Σ 0 K ¯ ∗ 0 , and $$ {\Xi}_c^0\to {\Sigma}^{+}{K}^{\ast -} $$ Ξ c 0 → Σ + K ∗ − for the first time. The relative branching ratios to the normalization mode of $$ {\Xi}_c^0\to {\Xi}^{-}{\pi}^{+} $$ Ξ c 0 → Ξ − π + are measured to be $$ {\displaystyle \begin{array}{c}\mathcal{B}\left({\Xi}_c^0\to \Lambda {\overline{K}}^{\ast 0}\right)/\mathcal{B}\left({\Xi}_c^0\to {\Xi}^{-}{\pi}^{+}\right)=0.18\pm 0.02\left(\mathrm{stat}.\right)\pm 0.01\left(\mathrm{syst}.\right),\\ {}\mathcal{B}\left({\Xi}_c^0\to {\Sigma}^0{\overline{K}}^{\ast 0}\right)/\mathcal{B}\left({\Xi}_c^0\to {\Xi}^{-}{\pi}^{+}\right)=0.69\pm 0.03\left(\mathrm{stat}.\right)\pm 0.03\left(\mathrm{syst}.\right),\\ {}\mathcal{B}\left({\Xi}_c^0\to {\Sigma}^{+}{K}^{\ast -}\right)/\mathcal{B}\left({\Xi}_c^0\to {\Xi}^{-}{\pi}^{+}\right)=0.34\pm 0.06\left(\mathrm{stat}.\right)\pm 0.02\left(\mathrm{syst}.\right),\end{array}} $$ B Ξ c 0 → Λ K ¯ ∗ 0 / B Ξ c 0 → Ξ − π + = 0.18 ± 0.02 stat . ± 0.01 syst . , B Ξ c 0 → Σ 0 K ¯ ∗ 0 / B Ξ c 0 → Ξ − π + = 0.69 ± 0.03 stat . ± 0.03 syst . , B Ξ c 0 → Σ + K ∗ − / B Ξ c 0 → Ξ − π + = 0.34 ± 0.06 stat . ± 0.02 syst . , where the uncertainties are statistical and systematic, respectively. We obtain $$ {\displaystyle \begin{array}{c}\mathcal{B}\left({\Xi}_c^0\to \Lambda {\overline{K}}^{\ast 0}\right)=\left(3.3\pm 0.3\left(\mathrm{stat}.\right)\pm 0.2\left(\mathrm{syst}.\right)\pm 1.0\left(\mathrm{ref}.\right)\right)\times {10}^{-3},\\ {}\mathcal{B}\left({\Xi}_c^0\to {\Sigma}^0{\overline{K}}^{\ast 0}\right)=\left(12.4\pm 0.5\left(\mathrm{stat}.\right)\pm 0.5\left(\mathrm{syst}.\right)\pm 3.6\left(\mathrm{ref}.\right)\right)\times {10}^{-3},\\ {}\mathcal{B}\left({\Xi}_c^0\to {\Sigma}^{+}{K}^{\ast 0}\right)=\left(6.1\pm 1.0\left(\mathrm{stat}.\right)\pm 0.4\left(\mathrm{syst}.\right)\pm 1.8\left(\mathrm{ref}.\right)\right)\times {10}^{-3},\end{array}} $$ B Ξ c 0 → Λ K ¯ ∗ 0 = 3.3 ± 0.3 stat . ± 0.2 syst . ± 1.0 ref . × 10 − 3 , B Ξ c 0 → Σ 0 K ¯ ∗ 0 = 12.4 ± 0.5 stat . ± 0.5 syst . ± 3.6 ref . × 10 − 3 , B Ξ c 0 → Σ + K ∗ 0 = 6.1 ± 1.0 stat . ± 0.4 syst . ± 1.8 ref . × 10 − 3 , where the uncertainties are statistical, systematic, and from $$ \mathcal{B}\left({\Xi}_c^0\to {\Xi}^{-}{\pi}^{+}\right) $$ B Ξ c 0 → Ξ − π + , respectively. The asymmetry parameters $$ \alpha \left({\Xi}_c^0\to \Lambda {\overline{K}}^{\ast 0}\right) $$ α Ξ c 0 → Λ K ¯ ∗ 0 and $$ \alpha \left({\Xi}_c^0\to {\Sigma}^{+}{K}^{\ast -}\right) $$ α Ξ c 0 → Σ + K ∗ − are 0 . 15 ± 0 . 22(stat . ) ± 0 . 04(syst . ) and − 0 . 52 ± 0 . 30(stat . ) ± 0 . 02(syst . ), respectively, where the uncertainties are statistical followed by systematic. 

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5. Lagrangian skeleta and plane curve singularities

   https://doi.org/10.1007/s11784-022-00939-8  

   Casals, Roger ( June 2022 , Journal of Fixed Point Theory and Applications)

   Abstract We construct closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities. This yields closed Lagrangian skeleta for Weinstein pairs $$(\mathbb {C}^2,\Lambda )$$ ( C 2 , Λ ) and Weinstein 4-manifolds $$W(\Lambda )$$ W ( Λ ) associated to max-tb Legendrian representatives of algebraic links $$\Lambda \subseteq (\mathbb {S}^3,\xi _\text {st})$$ Λ ⊆ ( S 3 , ξ st ) . We provide computations of Legendrian and Weinstein invariants, and discuss the contact topological nature of the Fomin–Pylyavskyy–Shustin–Thurston cluster algebra associated to a singularity. Finally, we present a conjectural ADE-classification for Lagrangian fillings of certain Legendrian links and list some related problems. 

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