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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.
2. 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 levelmore »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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3. Quantum phase transition in a clean superconductor with repulsive dynamical interaction

   https://doi.org/10.1038/s41535-022-00457-3  

   Pimenov, Dimitri ; Chubukov, Andrey V. ( April 2022 , npj Quantum Materials)

   We consider a model of electrons at zero temperature, with a repulsive interaction which is a function of the energy transfer. Such an interaction can arise from the combination of electron–electron repulsion at high energies and the weaker electron–phonon attraction at low energies. As shown in previous works, superconductivity can develop despite the overall repulsion due to the energy dependence of the interaction, but the gap Δ(ω) must change sign at some (imaginary) frequencyω₀to counteract the repulsion. However, when the constant repulsive part of the interaction is increased, a quantum phase transition towards the normal state occurs. We show that, as the phase transition is approached, Δ andω₀must vanish in a correlated way such that$$1/| \log [{{\Delta }}(0)]| \sim {\omega }_{0}^{2}$$$1/\mid \log \left[\Delta \left(0\right)\right]\mid ~{\omega }_0^2$. We discuss the behavior of phase fluctuations near this transition and show that the correlation between Δ(0) andω₀locks the phase stiffness to a non-zero value.
4. Efficient Computation of Sequence Mappability

   https://doi.org/10.1007/s00453-022-00934-y  

   Charalampopoulos, Panagiotis ; Iliopoulos, Costas S. ; Kociumaka, Tomasz ; Pissis, Solon P. ; Radoszewski, Jakub ; Straszyński, Juliusz ( February 2022 , Algorithmica)

   Sequence mappability is an important task in genome resequencing. In the (k, m)-mappability problem, for a given sequenceTof lengthn, the goal is to compute a table whoseith entry is the number of indices$$j \ne i$$$j\ne i$such that the length-msubstrings ofTstarting at positionsiandjhave at mostkmismatches. Previous works on this problem focused on heuristics computing a rough approximation of the result or on the case of$$k=1$$$k=1$. We present several efficient algorithms for the general case of the problem. Our main result is an algorithm that, for$$k=O(1)$$$k=O\left(1\right)$, works in$$O(n)$$$O\left(n\right)$space and, with high probability, in$$O(n \cdot \min \{m^k,\log ^k n\})$$$O(n·min\{m^k,{log}^{k}n\})$time. Our algorithm requires a careful adaptation of thek-errata trees of Cole et al. [STOC 2004] to avoid multiple counting of pairs of substrings. Our technique can also be applied to solve the all-pairs Hamming distance problem introduced by Crochemore et al. [WABI 2017]. We further develop$$O(n^2)$$$O\left(n^2\right)$-time algorithms to computeall(k, m)-mappability tables for a fixedmand all$$k\in \{0,\ldots ,m\}$$$k\in \{0,\dots ,m\}$or a fixedkand all$$m\in \{k,\ldots ,n\}$$$m\in \{k,\dots ,n\}$. Finally, we show that, for$$k,m = \Theta (\log n)$$$k,m=\Theta (logn)$, the (k, m)-mappability problem cannot be solved in strongly subquadratic time unless the Strong Exponential Time Hypothesis fails. This is an improved and extended version of a paper presented at SPIRE 2018.
5. Hermitian–Yang–Mills Connections on Collapsing Elliptically Fibered K3 Surfaces

   https://doi.org/10.1007/s12220-021-00808-9  

   Datar, Ved ; Jacob, Adam ( January 2022 , The Journal of Geometric Analysis)

   Let$$X\rightarrow {{\mathbb {P}}}^1$$$X\to P^1$be an elliptically fiberedK3 surface, admitting a sequence$$\omega _{i}$$${\omega }_i$of Ricci-flat metrics collapsing the fibers. LetVbe a holomorphicSU(n) bundle overX, stable with respect to$$\omega _i$$${\omega }_i$. Given the corresponding sequence$$\Xi _i$$${\Xi }_i$of Hermitian–Yang–Mills connections onV, we prove that, ifEis a generic fiber, the restricted sequence$$\Xi _i|_{E}$$${\Xi }_i{|}_E$converges to a flat connection$$A_0$$$A_0$. Furthermore, if the restriction$$V|_E$$${V|}_E$is of the form$$\oplus _{j=1}^n{\mathcal {O}}_E(q_j-0)$$${\oplus }_{j=1}^n{O}_E(q_j-0)$forndistinct points$$q_j\in E$$$q_j\in E$, then these points uniquely determine$$A_0$$$A_0$.