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1. Irreducibility of Recombination Markov Chains in the Triangular Lattice

   Cannon, Sarah ( May 2023 , SIAM Conference on Applied and Computational Discrete Algorithms (ACDA23))

   Berry, Jonathan ; Shmoys, David ; Cowen, Lenore ; Naumann, Uwe (Ed.)

   In the United States, regions (such as states or counties) are frequently divided into districts for the purpose of electing representatives. How the districts are drawn can have a profound effect on who's elected, and drawing the districts to give an advantage to a certain group is known as gerrymandering. It can be surprisingly difficult to detect when gerrymandering is occurring, but one algorithmic method is to compare a current districting plan to a large number of randomly sampled plans to see whether it is an outlier. Recombination Markov chains are often used to do this random sampling: randomly choose two districts, consider their union, and split this union up in a new way. This approach works well in practice and has been widely used, including in litigation, but the theory behind it remains underdeveloped. For example, it's not known if recombination Markov chains are irreducible, that is, if recombination moves suffice to move from any districting plan to any other. Irreducibility of recombination Markov chains can be formulated as a graph problem: for a planar graph G, is the space of all partitions of G into κ connected subgraphs (κ districts) connected by recombination moves? While the answer is yes when districts can be as small as one vertex, this is not realistic in real-world settings where districts must have approximately balanced populations. Here we fix district sizes to be κ1 ± 1 vertices, κ2 ± 1 vertices,… for fixed κ1, κ2,…, a more realistic setting. We prove for arbitrarily large triangular regions in the triangular lattice, when there are three simply connected districts, recombination Markov chains are irreducible. This is the first proof of irreducibility under tight district size constraints for recombination Markov chains beyond small or trivial examples. The triangular lattice is the most natural setting in which to first consider such a question, as graphs representing states/regions are frequently triangulated. The proof uses a sweep-line argument, and there is hope it will generalize to more districts, triangulations satisfying mild additional conditions, and other redistricting Markov chains. 

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2. Tunable analog thermal material

   https://doi.org/10.1038/s41467-020-19909-0  

   Xu, Guoqiang ; Dong, Kaichen ; Li, Ying ; Li, Huagen ; Liu, Kaipeng ; Li, Longqiu ; Wu, Junqiao ; Qiu, Cheng-Wei ( November 2020 , Nature Communications)

   Naturally-occurring thermal materials usually possess specific thermal conductivity (κ), forming a digital set ofκvalues. Emerging thermal metamaterials have been deployed to realize effective thermal conductivities unattainable in natural materials. However, the effective thermal conductivities of such mixing-based thermal metamaterials are still in digital fashion, i.e., the effective conductivity remains discrete and static. Here, we report an analog thermal material whose effective conductivity can be in-situ tuned from near-zero to near-infinityκ. The proof-of-concept scheme consists of a spinning core made of uncured polydimethylsiloxane (PDMS) and fixed bilayer rings made of silicone grease and steel. Thanks to the spinning PDMS and its induced convective effects, we can mold the heat flow robustly with continuously changing and anisotropicκ. Our work enables a single functional thermal material to meet the challenging demands of flexible thermal manipulation. It also provides platforms to investigate heat transfer in systems with moving components.

    

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3. Almost Optimal Scaling of Reed-Muller Codes on BEC and BSC Channels

   Hassani, Hamed ; Kudekar, Shrinivas ; Ordentlich, Or ; Polyanskiy, Yury ; Urbanke, Rudiger ( June 2018 , 2018 IEEE Int. Symp. Inf. Theory (ISIT))

   Consider a binary linear code of length N, minimum distance dmin, transmission over the binary erasure channel with parameter 0 <  < 1 or the binary symmetric channel with parameter 0 <  < 1 2 , and block-MAP decoding. It was shown by Tillich and Zemor that in this case the error probability of the block-MAP decoder transitions “quickly” from δ to 1−δ for any δ > 0 if the minimum distance is large. In particular the width of the transition is of order O(1/ √ dmin). We strengthen this result by showing that under suitable conditions on the weight distribution of the code, the transition width can be as small as Θ(1/N 1 2 −κ ), for any κ > 0, even if the minimum distance of the code is not linear. This condition applies e.g., to Reed-Mueller codes. Since Θ(1/N 1 2 ) is the smallest transition possible for any code, we speak of “almost” optimal scaling. We emphasize that the width of the transition says nothing about the location of the transition. Therefore this result has no bearing on whether a code is capacity-achieving or not. As a second contribution, we present a new estimate on the derivative of the EXIT function, the proof of which is based on the Blowing-Up Lemma. 

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4. About the existence of warm H-rich pulsating white dwarfs

   https://doi.org/10.1051/0004-6361/201936346  

   Althaus, Leandro G. ; Córsico, Alejandro H. ; Uzundag, Murat ; Vučković, Maja ; Baran, Andrzej S. ; Bell, Keaton J. ; Camisassa, María E. ; Calcaferro, Leila M. ; De Gerónimo, Francisco C. ; Kepler, Souza Oliveira ; et al ( January 2020 , Astronomy & Astrophysics)

   Context. The possible existence of warm ( T eff  ∼ 19 000 K) pulsating DA white dwarf (WD) stars, hotter than ZZ Ceti stars, was predicted in theoretical studies more than 30 yr ago. These studies reported the occurrence of g -mode pulsational instabilities due to the κ mechanism acting in the partial ionization zone of He below the H envelope in models of DA WDs with very thin H envelopes ( M H / M ⋆  ≲ 10 −10 ). However, to date, no pulsating warm DA WD has been discovered, despite the varied theoretical and observational evidence suggesting that a fraction of WDs should be formed with a range of very low H content. Aims. We re-examine the pulsational predictions for such WDs on the basis of new full evolutionary sequences. We analyze all the warm DAs observed by the TESS satellite up to Sector 9 in order to search for the possible pulsational signal. Methods. We computed WD evolutionary sequences of masses 0.58 and 0.80 M ⊙ with H content in the range −14.5 ≲ log( M H / M ⋆ )≲ − 10, appropriate for the study of pulsational instability of warm DA WDs. Initial models were extracted from progenitors that were evolved through very late thermal pulses on the early cooling branch. We use LPCODE stellar code into which we have incorporated a new full-implicit treatment of time-dependent element diffusion to precisely model the H–He transition zone in evolving WD models with very low H content. The nonadiabatic pulsations of our warm DA WD models were computed in the effective temperature range of 30 000 − 10 000 K, focusing on ℓ = 1 g modes with periods in the range 50 − 1500 s. Results. We find that traces of H surviving the very late thermal pulse float to the surface, eventually forming thin, growing pure H envelopes and rather extended H–He transition zones. We find that such extended transition zones inhibit the excitation of g modes due to partial ionization of He below the H envelope. Only in the cases where the H–He transition is assumed much more abrupt than predicted by diffusion do models exhibit pulsational instability. In this case, instabilities are found only in WD models with H envelopes in the range of −14.5 ≲ log( M H / M ⋆ )≲ − 10 and at effective temperatures higher than those typical for ZZ Ceti stars, in agreement with previous studies. None of the 36 warm DAs observed so far by TESS satellite are found to pulsate. Conclusions. Our study suggests that the nondetection of pulsating warm DAs, if WDs with very thin H envelopes do exist, could be attributed to the presence of a smooth and extended H–He transition zone. This could be considered as indirect proof that element diffusion indeed operates in the interior of WDs. 

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5. Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

   https://doi.org/10.1007/s00440-021-01070-4  

   Miller, Jason ; Sheffield, Scott ; Werner, Wendelin ( June 2021 , Probability Theory and Related Fields)

   We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$for$$\kappa '$$${\kappa }^{\prime }$in (4, 8) that is drawn on an independent$$\gamma $$$\gamma$-LQG surface for$$\gamma ^2=16/\kappa '$$${\gamma }^2=16/{\kappa }^{\prime }$. The results are similar in flavor to the ones from our companion paper dealing with$$\hbox {CLE}_{\kappa }$$${\text{CLE}}_{\kappa }$for$$\kappa $$$\kappa$in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: In our paper entitled “CLE Percolations” described the law of interfaces obtained when coloring the loops of a$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$independently into two colors with respective probabilitiespand$$1-p$$$1-p$. This description was complete up to one missing parameter$$\rho $$$\rho$. The results of the present paper about CLE on LQG allow us to determine its value in terms ofpand$$\kappa '$$${\kappa }^{\prime }$. It shows in particular that$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$and$$\hbox {CLE}_{16/\kappa '}$$${\text{CLE}}_{16/{\kappa }^{\prime }}$are related via a continuum analog of the Edwards-Sokal coupling between$$\hbox {FK}_q$$${\text{FK}}_q$percolation and theq-state Potts model (which makes sense even for non-integerqbetween 1 and 4) if and only if$$q=4\cos ^2(4\pi / \kappa ')$$$q=4{cos}^2(4\pi /{\kappa }^{\prime })$. This provides further evidence for the long-standing belief that$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$and$$\hbox {CLE}_{16/\kappa '}$$${\text{CLE}}_{16/{\kappa }^{\prime }}$represent the scaling limits of$$\hbox {FK}_q$$${\text{FK}}_q$percolation and theq-Potts model whenqand$$\kappa '$$${\kappa }^{\prime }$are related in this way. Another consequence of the formula for$$\rho (p,\kappa ')$$$\rho (p,{\kappa }^{\prime })$is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

    

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