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1. The Vlasov–Poisson System with a Perfectly Conducting Wall: Convex Domains

   https://doi.org/10.1007/s00220-026-05731-0  

   Huang, Wenrui; Pausader, Benoit; Suzuki, Masahiro (October 2026, Communications in Mathematical Physics)

   We consider the Vlasov–Poisson system in a convex domain D with a perfectly conducting wall. We introduce the asymptotic domain for the domain D. Then under acceptable assumptions on D, we show that for localized initial data, the velocity of particles is asymptotically supported in the (closure of the) asymptotic domain and the solutions exhibit the asymptotics of modified scattering. 

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2. Nadaraya–Watson kernel smoothing as a random energy model

   https://doi.org/10.1088/1742-5468/ada49a  

   Zavatone-Veth, Jacob A; Pehlevan, Cengiz (January 2025, Journal of Statistical Mechanics: Theory and Experiment)

   Abstract Precise asymptotics have revealed many surprises in high-dimensional regression. These advances, however, have not extended to perhaps the simplest estimator: direct Nadaraya–Watson (NW) kernel smoothing. Here, we describe how one can use ideas from the analysis of the random energy model (REM) in statistical physics to compute sharp asymptotics for the NW estimator when the sample size is exponential in the dimension. As a simple starting point for investigation, we focus on the case in which one aims to estimate a single-index target function using a radial basis function kernel on the sphere. Our main result is a pointwise asymptotic for the NW predictor, showing that it re-scales the argument of the true link function. Our work provides a first step towards a detailed understanding of kernel smoothing in high dimensions. 

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3. Asymptotics of singular values for quantum derivatives

   https://doi.org/10.1090/tran/8827  

   Frank, Rupert; Sukochev, Fedor; Zanin, Dmitriy (March 2023, Transactions of the American Mathematical Society)

   We obtain Weyl type asymptotics for the quantised derivative \dj \mkern 1muf of a function f f from the homgeneous Sobolev space W ˙ d 1 ( R d ) \dot {W}^1_d(\mathbb {R}^d) on R d . \mathbb {R}^d. The asymptotic coefficient ‖ ∇ f ‖ L d ( R d ) \|\nabla f\|_{L_d(\mathbb R^d)} is equivalent to the norm of \dj \mkern 1muf in the principal ideal L d , ∞ , \mathcal {L}_{d,\infty }, thus, providing a non-asymptotic, uniform bound on the spectrum of \dj \mkern 1muf. Our methods are based on the C ∗ C^{\ast } -algebraic notion of the principal symbol mapping on R d \mathbb {R}^d , as developed recently by the last two authors and collaborators. 

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4. Long‐time asymptotics and the radiation condition with time‐periodic boundary conditions for linear evolution equations on the half‐line and experiment

   https://doi.org/10.1111/sapm.12665  

   Mao, Yifeng; Mantzavinos, Dionyssios; Hoefer, Mark A (April 2024, Studies in Applied Mathematics)

   Abstract The asymptotic Dirichlet‐to‐Neumann (D‐N) map is constructed for a class of scalar, constant coefficient, linear, third‐order, dispersive equations with asymptotically time/periodic Dirichlet boundary data and zero initial data on the half‐line, modeling a wavemaker acting upon an initially quiescent medium. The large time asymptotics for the special cases of the linear Korteweg‐de Vries and linear Benjamin–Bona–Mahony (BBM) equations are obtained. The D‐N map is proven to be unique if and only if the radiation condition that selects the unique wave number branch of the dispersion relation for a sinusoidal, time‐dependent boundary condition holds: (i) for frequencies in a finite interval, the wave number is real and corresponds to positive group velocity, and (ii) for frequencies outside the interval, the wave number is complex with positive imaginary part. For fixed spatial location , the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time‐periodic wave. The linearized BBM asymptotics are found to quantitatively agree with viscous core‐annular fluid experiments. 

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5. Holographic a-functions and Boomerang RG flows

   https://doi.org/10.1007/JHEP02(2024)019  

   Cáceres, Elena; Vásquez, Rodrigo Castillo; Landsteiner, Karl; Landea, Ignacio Salazar (February 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We use the radial null energy condition to construct a monotonica-function for a certain type of non-relativistic holographic RG flows. We test oura-function in three different geometries that feature a Boomerang RG flow, characterized by a domain wall between two AdS spaces with the same AdS radius, but with different (and sometimes direction-dependent) speeds of light. We find that thea-function monotonically decreases and goes to a constant in the asymptotic regimes of the geometry. Using the holographic dictionary in this asymptotic AdS spaces, we find that thea-function not only reads the fixed point central charge but also the speed of light, suggesting what the correct RG charge might be for non-relativistic RG flows. 

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