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1. Asymptotics of noncolliding q-exchangeable random walks

   https://doi.org/10.1088/1751-8121/acedda  

   Petrov, Leonid ; Tikhonov, Mikhail ( August 2023 , Journal of Physics A: Mathematical and Theoretical)

   We consider a process of noncollidingq-exchangeable random walks on$\mathbb{Z}$making steps 0 (‘straight’) and −1 (‘down’). A single random walk is calledq-exchangeable if under an elementary transposition of the neighboring steps$(\text{down},\text{straight})\to (\text{straight},\text{down})$the probability of the trajectory is multiplied by a parameter$q\in (0,1)$. Our process ofmnoncollidingq-exchangeable random walks is obtained from the independentq-exchangeable walks via the Doob’sh-transform for a nonnegative eigenfunctionh(expressed via theq-Vandermonde product) with the eigenvalue less than 1. The system ofmwalks evolves in the presence of an absorbing wall at 0. The repulsion mechanism is theq-analogue of the Coulomb repulsion of random matrix eigenvalues undergoing Dyson Brownian motion. However, in our model, the particles are confined to the positive half-line and do not spread as Brownian motions or simple random walks. We show that the trajectory of the noncollidingq-exchangeable walks started from an arbitrary initial configuration forms a determinantal point process, and express its kernel in a double contour integral form. This kernel is obtained as a limit from the correlation kernel ofq-distributed random lozenge tilings of sawtooth polygons. In the limit as$m\to \mathrm{\infty }$,$q=e^{-\gamma /m}$withγ > 0 fixed, and under a suitable scaling of the initial data, we obtain a limit shape of our noncolliding walks and also show that their local statistics are governed by the incomplete beta kernel. The latter is a distinguished translation invariant ergodic extension of the two-dimensional discrete sine kernel.

    

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2. Global existence and singularity formation for the generalized Constantin–Lax–Majda equation with dissipation: the real line vs. periodic domains

   https://doi.org/10.1088/1361-6544/ad140c  

   Ambrose, David M ; Lushnikov, Pavel M ; Siegel, Michael ; Silantyev, Denis A ( December 2023 , Nonlinearity)

   The question of global existence versus finite-time singularity formation is considered for the generalized Constantin–Lax–Majda equation with dissipation$-{\mathrm{\Lambda }}^{\sigma }$, where$\widehat{{\mathrm{\Lambda }}^{\sigma }}=|k{|}^{\sigma }$, both for the problem on the circle$x\in [-\pi ,\pi ]$and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for$\sigma \text{⩾}1$and all real values of an advection parameterawhen the data is small. We also derive new analytical solutions in both geometries whena = 0, and on the real line when$a=1/2$, for various values ofσ. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet fora = 0 andσ = 2, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values ofσthat are greater than or equal to one, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion of singularities in the complex plane. The computations validate and build upon the analytical theory.

    

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3. Imaging topology of Hofstadter ribbons

   https://doi.org/10.1088/1367-2630/ab165b  

   Genkina, Dina ; Aycock, Lauren M. ; Lu, Hsin-I ; Lu, Mingwu ; Pineiro, Alina M. ; Spielman, I. B. ( May 2019 , New Journal of Physics)

   Physical systems with non-trivial topological order find direct applications in metrology (Klitzinget al1980Phys.Rev. Lett.45494–7) and promise future applications in quantum computing (Freedman 2001Found. Comput. Math.1183–204; Kitaev 2003Ann. Phys.3032–30). The quantum Hall effect derives from transverse conductance, quantized to unprecedented precision in accordance with the system’s topology (Laughlin 1981Phys. Rev.B235632–33). At magnetic fields beyond the reach of current condensed matter experiment, around${\mathrm{10}}^{\mathrm{4}}$T, this conductance remains precisely quantized with values based on the topological order (Thoulesset al1982Phys. Rev. Lett.49405–8). Hitherto, quantized conductance has only been measured in extended 2D systems. Here, we experimentally studied narrow 2D ribbons, just 3 or 5 sites wide along one direction, using ultracold neutral atoms where such large magnetic fields can be engineered (Jaksch and Zoller 2003New J. Phys.556; Miyakeet al2013Phys. Rev. Lett.111185302; Aidelsburgeret al2013Phys. Rev. Lett.111185301; Celiet al2014Phys. Rev. Lett.112043001; Stuhlet al2015Science3491514; Manciniet al2015Science3491510; Anet al2017Sci. Adv.3). We microscopically imaged the transverse spatial motion underlying the quantized Hall effect. Our measurements identify the topological Chern numbers with typical uncertainty of$\mathrm{5}\%$, and show that although band topology is only properly defined in infinite systems, its signatures are striking even in nearly vanishingly thin systems.

    

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4. Haake–Lewenstein–Wilkens approach to spin-glasses revisited

   https://doi.org/10.1088/1751-8121/ac9d10  

   Lewenstein, Maciej ; Cirauqui, David ; Ángel García-March, Miguel ; Guigó i Corominas, Guillem ; Grzybowski, Przemysław ; Saavedra, José R ; Wilkens, Martin ; Wehr, Jan ( November 2022 , Journal of Physics A: Mathematical and Theoretical)

   We revisit the Haake–Lewenstein–Wilkens approach to Edwards–Anderson (EA) model of Ising spin glass (SG) (Haakeet al1985Phys. Rev. Lett.552606). This approach consists in evaluation and analysis of the probability distribution of configurations of two replicas of the system, averaged over quenched disorder. This probability distribution generates squares of thermal copies of spin variables from the two copies of the systems, averaged over disorder, that is the terms that enter the standard definition of the original EA order parameter,$q_{\text{EA}}$. We use saddle point/steepest descent (SPSD) method to calculate the average of the Gaussian disorder in higher dimensions. This approximate result suggest that$q_{\text{EA}}>0$at$0<T<T_c$in 3D and 4D. The case of 2D seems to be a little more subtle, since in the present approach energy increase for a domain wall competes with boundary/edge effects more strongly in 2D; still our approach predicts SG order at sufficiently low temperature. We speculate, how these predictions confirm/contradict widely spread opinions that: (i) There exist only one (up to the spin flip) ground state in EA model in 2D, 3D and 4D; (ii) there is (no) SG transition in 3D and 4D (2D). This paper is dedicated to the memories of Fritz Haake and Marek Cieplak.

    

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5. Deciphering the sensitivity of urban canopy air temperature to anthropogenic heat flux with a forcing-feedback framework

   https://doi.org/10.1088/1748-9326/ace7e0  

   Wang, Linying ; Sun, Ting ; Zhou, Wenyu ; Liu, Maofeng ; Li, Dan ( August 2023 , Environmental Research Letters)

   The sensitivity of urban canopy air temperature ($T_a$) to anthropogenic heat flux ($Q_{AH}$) is known to vary with space and time, but the key factors controlling such spatiotemporal variabilities remain elusive. To quantify the contributions of different physical processes to the magnitude and variability of$\mathrm{\Delta }{T}_a/\mathrm{\Delta }{Q}_{AH}$(where$\mathrm{\Delta }$represents a change), we develop a forcing-feedback framework based on the energy budget of air within the urban canopy layer and apply it to diagnosing$\mathrm{\Delta }{T}_a/\mathrm{\Delta }{Q}_{AH}$simulated by the Community Land Model Urban over the contiguous United States (CONUS). In summer, the median$\mathrm{\Delta }{T}_a/\mathrm{\Delta }{Q}_{AH}$is around 0.01$\text{K\hspace{0.17em}}{\left(\text{W\hspace{0.17em}}{\text{m}}^{-\text{2}}\right)}^{-1}$over the CONUS. Besides the direct effect of$Q_{AH}$on$T_a$, there are important feedbacks through changes in the surface temperature, the atmosphere–canopy air heat conductance ($c_a$), and the surface–canopy air heat conductance. The positive and negative feedbacks nearly cancel each other out and$\mathrm{\Delta }{T}_a/\mathrm{\Delta }{Q}_{AH}$is mostly controlled by the direct effect in summer. In winter,$\mathrm{\Delta }{T}_a/\mathrm{\Delta }{Q}_{AH}$becomes stronger, with the median value increased by about 20% due to weakened negative feedback associated with$c_a$. The spatial and temporal (both seasonal and diurnal) variability of$\mathrm{\Delta }{T}_a/\mathrm{\Delta }{Q}_{AH}$as well as the nonlinear response of$\mathrm{\Delta }{T}_a$to$\mathrm{\Delta }{Q}_{AH}$are strongly related to the variability of$c_a$, highlighting the importance of correctly parameterizing convective heat transfer in urban canopy models.

    

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