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1. Stability and chaos in dynamical last passage percolation

   https://doi.org/10.1090/cams/35  

   Ganguly, Shirshendu; Hammond, Alan (June 2024, Communications of the American Mathematical Society)

   Many complex disordered systems in statistical mechanics are characterized by intricate energy landscapes. The ground state, the configuration with lowest energy, lies at the base of the deepest valley. In important examples, such as Gaussian polymers and spin glass models, the landscape has many valleys and the abundance of near-ground states (at the base of valleys) indicates the phenomenon ofchaos, under which the ground state alters profoundly when the disorder of the model is slightly perturbed. In this article, we compute the critical exponent that governs the onset of chaos in a dynamic manifestation of a canonical model in the Kardar-Parisi-Zhang [KPZ] universality class, Brownian last passage percolation [LPP]. In this model in its static form, semidiscrete polymers advance through Brownian noise, their energy given by the integral of the white noise encountered along their journey. A ground state is ageodesic, of extremal energy given its endpoints. We perturb Brownian LPP by evolving the disorder under an Ornstein-Uhlenbeck flow. We prove that, for polymers of length $n$, a sharp phase transition marking the onset of chaos is witnessed at the critical time $n^{-<\#comment/>1/3}$. Indeed, the overlap between the geodesics at times zero and $t>0$that travel a given distance of order $n$will be shown to be of order $n$when $t\ll <\#comment/>n^{-<\#comment/>1/3}$; and to be of smaller order when $t\gg <\#comment/>n^{-<\#comment/>1/3}$. We expect this exponent to be universal across a wide range of interface models. The present work thus sheds light on the dynamical aspect of the KPZ class; it builds on several recent advances. These include Chatterjee’s harmonic analytic theory [Superconcentration and related topics, Springer, Cham, 2014] of equivalence ofsuperconcentrationandchaosin Gaussian spaces; a refined understanding of the static landscape geometry of Brownian LPP developed in the companion paper (see S. Ganguly and A. Hammond [Electron. J. Probab. 28 (2023), 80 pp.]); and, underlying the latter, strong comparison estimates of the geodesic energy profile to Brownian motion (see J. Calvert, A. Hammond, and M. Hegde [Astérisque 441 (2023), pp. v+119]). 

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2. Characterization of positive definite, radial functions on free groups

   https://doi.org/10.1090/bproc/158  

   Chuah, Chian Yeong; Liu, Zhenchuan; Mei, Tao (April 2023, Proceedings of the American Mathematical Society, Series B)

   This article studies the properties of positive definite, radial functions on free groups following the work of Haagerup and Knudby [Proc. Amer. Math. Soc. 143 (2015), pp. 1477–1489]. We obtain characterizations of radial functions with respect to the ${\mathrm{\ell <\#comment/>}}^2$length on the free groups with infinite generators and the characterization of the positive definite, radial functions with respect to the ${\mathrm{\ell <\#comment/>}}^p$length on the free real line with infinite generators for $0>p\le <\#comment/>2$. We obtain a Lévy-Khintchine formula for length-radial conditionally negative functions as well. 

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3. Super-polynomial accuracy of one dimensional randomized nets using the median of means

   https://doi.org/10.1090/mcom/3791  

   Pan, Zexin; Owen, Art (March 2023, Mathematics of Computation)

   Let $f$be analytic on $[0,1]$with $|f^{\left(k\right)}\left(1/2\right)|⩽<\#comment/>A{\mathrm{\alpha <\#comment/>}}^{k}k!$for some constants $A$and $\mathrm{\alpha <\#comment/>}>2$and all $k⩾<\#comment/>1$. We show that the median estimate of $\mathrm{\mu <\#comment/>}={\int <\#comment/>}_0^{1}f\left(x\right)\mathrm{d}x$under random linear scrambling with $n=2^m$points converges at the rate $O\left(n^{-<\#comment/>c\log <\#comment/>\left(n\right)}\right)$for any $c>3\log <\#comment/>\left(2\right)/{\mathrm{\pi <\#comment/>}}^2\approx <\#comment/>0.21$. We also get a super-polynomial convergence rate for the sample median of $2k-<\#comment/>1$random linearly scrambled estimates, when $k/m$is bounded away from zero. When $f$has a $p$’th derivative that satisfies a $\mathrm{\lambda <\#comment/>}$-Hölder condition then the median of means has error $O\left(n^{-<\#comment/>(p+\mathrm{\lambda <\#comment/>})+\mathrm{ϵ<\#comment/>}}\right)$for any $\mathrm{ϵ<\#comment/>}>0$, if $k\to <\#comment/>\mathrm{\infty <\#comment/>}$as $m\to <\#comment/>\mathrm{\infty <\#comment/>}$. The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number. 

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4. An optimal approximation problem for free polynomials

   https://doi.org/10.1090/proc/16474  

   Arora, Palak; Augat, Meric; Jury, Michael; Sargent, Meredith (November 2023, Proceedings of the American Mathematical Society)

   Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial $f$in $d$freely noncommuting arguments, find a free polynomial $p_n$, of degree at most $n$, to minimize $c_n≔\Vert <\#comment/>p_{n}f-<\#comment/>1{\Vert <\#comment/>}^2$. (Here the norm is the ${\mathrm{\ell <\#comment/>}}^2$norm on coefficients.) We show that $c_n\to <\#comment/>0$if and only if $f$is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the $d$-shift. 

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5. A nonlinear least-squares convexity enforcing 𝐶⁰ interior penalty method for the Monge–Ampère equation on strictly convex smooth planar domains

   https://doi.org/10.1090/cams/39  

   Brenner, Susanne; Sung, Li-yeng; Tan, Zhiyu; Zhang, Hongchao (October 2024, Communications of the American Mathematical Society)

   We construct a nonlinear least-squares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the Monge-Ampère equation on strictly convex smooth domains in ${\mathbb{R}}^2$. It is based on an isoparametric $C^0$finite element space with exotic degrees of freedom that can enforce the convexity of the approximate solutions.A priorianda posteriorierror estimates together with corroborating numerical results are presented. 

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