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1. The Brown measure of the free multiplicative Brownian motion

   https://doi.org/10.1007/s00440-022-01142-z  

   Driver, Bruce K. ; Hall, Brian ; Kemp, Todd ( October 2022 , Probability Theory and Related Fields)

   The free multiplicative Brownian motion$$b_{t}$$$b_t$is the large-Nlimit of the Brownian motion on$$\mathsf {GL}(N;\mathbb {C}),$$$\mathrm{GL}(N;C),$in the sense of$$*$$$\ast$-distributions. The natural candidate for the large-Nlimit of the empirical distribution of eigenvalues is thus the Brown measure of$$b_{t}$$$b_t$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region$$\Sigma _{t}$$${\Sigma }_t$that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density$$W_{t}$$$W_t$on$$\overline{\Sigma }_{t},$$${\overline{\Sigma }}_t,$which is strictly positive and real analytic on$$\Sigma _{t}$$${\Sigma }_t$. This density has a simple form in polar coordinates:$$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$$\begin{array}{c}{W}_t(r,\theta )=\frac{1}{r^2}{w}_t\left(\theta \right),\end{array}$where$$w_{t}$$$w_t$is an analytic function determined by the geometry of the region$$\Sigma _{t}$$${\Sigma }_t$. We show also that the spectral measure of free unitary Brownian motion$$u_{t}$$$u_t$is a “shadow” of the Brown measure of$$b_{t}$$$b_t$, precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.

    

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2. New upper bounds for spherical codes and packings

   https://doi.org/10.1007/s00208-023-02738-z  

   Sardari, Naser Talebizadeh ; Zargar, Masoud ( October 2023 , Mathematische Annalen)

   We improve the previously best known upper bounds on the sizes of$$\theta $$$\theta$-spherical codes for every$$\theta <\theta ^*\approx 62.997^{\circ }$$$\theta <{\theta }^{\ast }\approx 62.{997}^{\circ }$at least by a factor of 0.4325, in sufficiently high dimensions. Furthermore, for sphere packing densities in dimensions$$n\ge 2000$$$n\ge 2000$we have an improvement at least by a factor of$$0.4325+\frac{51}{n}$$$0.4325+\frac{51}{n}$. Our method also breaks many non-numerical sphere packing density bounds in smaller dimensions. This is the first such improvement for each dimension since the work of Kabatyanskii and Levenshtein (Problemy Peredači Informacii 14(1):3–25, 1978) and its later improvement by Levenshtein (Dokl Akad Nauk SSSR 245(6):1299–1303, 1979) . Novelties of this paper include the analysis of triple correlations, usage of the concentration of mass in high dimensions, and the study of the spacings between the roots of Jacobi polynomials.

    

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3. 𝐿₁-distortion of Wasserstein metrics: A tale of two dimensions

   https://doi.org/10.1090/btran/143  

   Baudier, F. ; Gartland, C. ; Schlumprecht, Th. ( August 2023 , Transactions of the American Mathematical Society, Series B)

   By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid$\{0,1,\dots , n\}^2$has$L_1$-distortion bounded below by a constant multiple of$\sqrt {\log n}$. We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if$\{G_n\}_{n=1}^\infty$is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number$\delta \in [2,\infty )$, then the 1-Wasserstein metric over$G_n$has$L_1$-distortion bounded below by a constant multiple of$(\log |G_n|)^{\frac {1}{\delta }}$. We proceed to compute these dimensions for$\oslash$-powers of certain graphs. In particular, we get that the sequence of diamond graphs$\{\mathsf {D}_n\}_{n=1}^\infty$has isoperimetric dimension and Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the 1-Wasserstein metric over$\mathsf {D}_n$has$L_1$-distortion bounded below by a constant multiple of$\sqrt {\log | \mathsf {D}_n|}$. This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of$L_1$-embeddable graphs whose sequence of 1-Wasserstein metrics is not$L_1$-embeddable.

    

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4. 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 level 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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5. Enumeration of Rooted Binary Unlabeled Galled Trees

   https://doi.org/10.1007/s11538-024-01270-8  

   Agranat-Tamir, Lily ; Mathur, Shaili ; Rosenberg, Noah A. ( March 2024 , Bulletin of Mathematical Biology)

   Rooted binarygalledtrees generalize rooted binary trees to allow a restricted class of cycles, known asgalls. We build upon the Wedderburn-Etherington enumeration of rooted binary unlabeled trees withnleaves to enumerate rooted binary unlabeled galled trees withnleaves, also enumerating rooted binary unlabeled galled trees withnleaves andggalls,$$0 \leqslant g \leqslant \lfloor \frac{n-1}{2} \rfloor $$$0⩽g⩽\lfloor \frac{n-1}{2}\rfloor$. The enumerations rely on a recursive decomposition that considers subtrees descended from the nodes of a gall, adopting a restriction on galls that amounts to considering only the rooted binarynormalunlabeled galled trees in our enumeration. We write an implicit expression for the generating function encoding the numbers of trees for alln. We show that the number of rooted binary unlabeled galled trees grows with$$0.0779(4.8230^n)n^{-\frac{3}{2}}$$$0.0779(4.{8230}^n)n^{-\frac{3}{2}}$, exceeding the growth$$0.3188(2.4833^n)n^{-\frac{3}{2}}$$$0.3188(2.{4833}^n)n^{-\frac{3}{2}}$of the number of rooted binary unlabeled trees without galls. However, the growth of the number of galled trees with only one gall has the same exponential order 2.4833 as the number with no galls, exceeding it only in the subexponential term,$$0.3910n^{\frac{1}{2}}$$$0.3910n^{\frac{1}{2}}$compared to$$0.3188n^{-\frac{3}{2}}$$$0.3188n^{-\frac{3}{2}}$. For a fixed number of leavesn, the number of gallsgthat produces the largest number of rooted binary unlabeled galled trees lies intermediate between the minimum of$$g=0$$$g=0$and the maximum of$$g=\lfloor \frac{n-1}{2} \rfloor $$$g=\lfloor \frac{n-1}{2}\rfloor$. We discuss implications in mathematical phylogenetics.

    

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