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Spectral Independence in High-Dimensional Expanders and Applications to the Hardcore Model [article]

Nima Anari, Kuikui Liu, Shayan Oveis Gharan
2020 arXiv   pre-print
As an application, we show that natural Glauber dynamics mixes rapidly (in polynomial time) to generate a random independent set from the hardcore model up to the uniqueness threshold.  ...  We prove that if μ is spectrally independent, then the corresponding high dimensional simplicial complex is a local spectral expander.  ...  Connections to High Dimensional Simplicial Complexes Let us first phrase our main contribution in the language of high-dimensional expanders.  ... 
arXiv:2001.00303v3 fatcat:g4gzn2kobbgkhmbnff7qhgvawm

Spectral independence, coupling with the stationary distribution, and the spectral gap of the Glauber dynamics [article]

Vishesh Jain, Huy Tuan Pham, Thuy Duong Vuong
2021 arXiv   pre-print
regular graph G = (V,E) with degree Δ≥ 3 and girth at least 6, and for any ε, δ > 0, the partition function of the hardcore model with fugacity λ≤ (1-δ)λ_c(Δ) may be approximated within a (1+ε)-multiplicative  ...  As concrete applications, by combining our new lower bound with known spectral independence computations and known coupling arguments: (1) We show that for a triangle-free graph G = (V,E) with maximum  ...  Notably, in the case of well-studied spin systems such as uniform k-colorings of triangle-free graphs and the hardcore model on high girth graphs, where coupling methods have succeeded in analysing the  ... 
arXiv:2105.01201v1 fatcat:rbdzejscbfdqncozjimkidsbte

Entropic Independence II: Optimal Sampling and Concentration via Restricted Modified Log-Sobolev Inequalities [article]

Nima Anari, Vishesh Jain, Frederic Koehler, Huy Tuan Pham, Thuy-Duong Vuong
2021 arXiv   pre-print
We show how to sample from the hardcore and Ising models on n-node graphs that have a constant δ relative gap to the tree-uniqueness threshold, in nearly-linear time O_δ(n).  ...  To establish restricted entropy contraction, we extend the entropic independence framework of Anari, Jain, Koehler, Pham, and Vuong to the setting of distributions that are spectrally independent under  ...  A sequence of results following the relatively new approach of high-dimensional-expanders-based Markov chain analysis, recently achieved the near-linear mixing time goal with no structural assumption other  ... 
arXiv:2111.03247v1 fatcat:wxayy7gnavdubgxknmrp5k3m7u

Hard-core boson approach to thespin−12triangular-lattice antiferromagnetCs2CuCl4at finite temperatures in magnetic fields higher than the saturation field

Simon Streib, Peter Kopietz
2015 Physical Review B  
We study the high magnetic field regime of the antiferromagnetic insulator Cs_2CuCl_4 by expressing the spin-1/2 operators in the relevant Heisenberg model in terms of hard-core bosons and implementing  ...  We calculate the spectral function of the hard-core bosons from which we obtain the in-plane components of the dynamic structure factor, the magnetic susceptibility, and the specific heat.  ...  ladder approximation even extends the applicability to arbitrarily high temperatures. 30 In this work, we apply the self-consistent ladder approximation to the relevant two-dimensional model for Cs  ... 
doi:10.1103/physrevb.92.094442 fatcat:6sed4zerw5h3nnozfijpvoqeia

A comparison of functional summary statistics to detect anisotropy of three-dimensional point patterns [article]

Farzaneh Safavimanesh, Claudia Redenbach
2016 arXiv   pre-print
In this paper, we focus on two recently developed summary statistics, the conical and the cylindrical K-function, which may be used to detect anisotropies in 3D point patterns.  ...  The growing availability of three-dimensional point process data asks for a development of suitable analysis techniques.  ...  Stochastic Geometry and Advanced Bioimaging, funded by a grant from the Villum Foundation, and by the Center for Mathematical and Computational Modelling (CM) 2 funded by the state of Rhineland-Palatinate  ... 
arXiv:1604.04211v1 fatcat:tzitn53stfhvtnzrlmusbxzj5y

Finite-temperature line shapes of hard-core bosons in quantum magnets: A diagrammatic approach tested in one dimension

Benedikt Fauseweh, Joachim Stolze, Götz S. Uhrig
2014 Physical Review B  
Thereby, a flexible approach is introduced which can also be applied to more sophisticated and higher dimensional models.  ...  Conceptually, the approach is not restricted to one dimension, but the one-dimensional case offers the opportunity to gauge the method by comparison to exact results obtained via a mapping to Jordan-Wigner  ...  B.F. acknowledges the Fakultät Physik of the Technische Universität Dortmund for his funding in the program "Bestenförderung".  ... 
doi:10.1103/physrevb.90.024428 fatcat:oywluhjotbhcxcm24ppeckr2dm

From Coupling to Spectral Independence and Blackbox Comparison with the Down-Up Walk [article]

Kuikui Liu
2021 arXiv   pre-print
To achieve this, we use Stein's method for Markov chains to show that a "good" coupling for a local Markov chain yields strong bounds on the spectral independence of the distribution in the sense of .  ...  Our primary application is to sampling proper list-colorings on bounded-degree graphs.  ...  We eschew the use of much of the terminology of high-dimensional expanders so as to simplify the exposition.  ... 
arXiv:2103.11609v2 fatcat:mxgh3owfx5fn3jk24x2f5pegsm

Effects of interactions on dynamic correlations of hard-core bosons at finite temperatures

Benedikt Fauseweh, Götz S. Uhrig
2017 Physical review B  
In order to include these effects quantitatively we extend the previously developed Br\"uckner approach to include also nearest-neighbor (NN) and next-nearest neighbor (NNN) interactions correctly in a  ...  Exemplarily, we consider the Heisenberg spin ladder to illustrate the significance of the additional interactions on the spectral functions at finite temperature which are proportional to inelastic neutron  ...  The Hartree and Fock contribution of the pure hard-core repulsion reads Σ Hart,U = Σ Fock,U = 1 N k ∞ −∞ dx ′ A k (x ′ )U 1 e βx ′ − 1 . (23) Expanding the high energy Hartree term in 1/U and combining  ... 
doi:10.1103/physrevb.96.115150 fatcat:fnpfeo7m7bfjhprate5cpylbaa

Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion [article]

Zongchen Chen, Kuikui Liu, Eric Vigoda
2021 arXiv   pre-print
As an application of our results, for the hard-core model on independent sets weighted by a fugacity λ, we establish O(nlogn) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum  ...  Our work presents an improved version of the spectral independence approach of Anari et al. (2020) and shows O(nlogn) mixing time on any n-vertex graph of bounded degree when the maximum eigenvalue of  ...  In contrast, previous works utilizing the spectral independence approach [ALO20 ; CLV20] and related works on high-dimensional expanders [KM17; DK17; Opp18; KO20; AL20] consider the spectral gap (or  ... 
arXiv:2011.02075v3 fatcat:pji4jik7mvdyldw4re67gzy6ky

Free coherent spinons in quantum square ice

Stefanos Kourtis, Claudio Castelnovo
2016 Physical review B  
We obtain exact results for excitation dynamics in both analytically solvable effective models and a fully interacting model of quantum square ice on finite clusters.  ...  We investigate the quantum dynamics of monopole-like excitations in quantum square ice, as captured by the strongly anisotropic spin-1/2 XXZ model on the checkerboard lattice.  ...  ACKNOWLEDGMENTS The authors are grateful to B. Douçot, D. L. Kovrizhin, R. Moessner  ... 
doi:10.1103/physrevb.94.104401 fatcat:g2vadrnnurb5xiejrwh2hok2eq

Fully-programmable universal quantum simulator with a one-dimensional quantum processor [article]

V. M. Bastidas, T. Haug, C. Gravel, L.-C. Kwek, W. J. Munro, Kae Nemoto
2020 arXiv   pre-print
For practical applications it is highly desirable to reconfigure the connectivity of the device, which for superconducting quantum processors is determined at fabrication.  ...  In addition, we require a careful design of control lines and couplings to resonators for measurements.  ...  Therefore, given the high degree of control the current technology already allows, it is now necessary to design a pathway towards practical applications of such devices.  ... 
arXiv:2009.00823v1 fatcat:gyusmwdqlvednkfglbrc7s2h3a

Hypercontractivity on High Dimensional Expanders: Approximate Efron-Stein Decompositions for ε-Product Spaces [article]

Tom Gur, Noam Lifshitz, Siqi Liu
2021 arXiv   pre-print
As applications, we obtain Fourier concentration, small-set expansion, and Kruskal-Katona theorems for high dimensional expanders.  ...  We prove hypercontractive inequalities on high dimensional expanders.  ...  Spectral independence in high-dimensional expanders and appli- cations to the hardcore model. SIAM Journal on Computing, (0):FOCS20–1, 2021.  ... 
arXiv:2111.09375v4 fatcat:pd4s5ihsqjcdvox5sikcrqmaoy

Exact thresholds for Ising–Gibbs samplers on general graphs

Elchanan Mossel, Allan Sly
2013 Annals of Probability  
Both results are tight, as it is known that the mixing time for random regular and Erdos-Renyi random graphs is, with high probability, exponential in n when (d-1)β>1, and dβ>1, respectively.  ...  We show that if (d-1)β<1, then there exists a constant C such that the discrete time mixing time of Gibbs samplers for the ferromagnetic Ising model on any graph of n vertices and maximal degree d, where  ...  Sly was a student at UC Berkeley and on visits to Rome Tre and the Weizmann Institute. A. Sly would like to thank Fabio Martinelli for useful discussions.  ... 
doi:10.1214/11-aop737 fatcat:5gytp3ht4res5ijsfgla26pcka

Quasiparticles as Detector of Topological Quantum Phase Transitions [article]

Sourav Manna, N. S. Srivatsa, Julia Wildeboer, Anne E. B. Nielsen
2020 arXiv   pre-print
They apply under different conditions, but do not cover the full range of many-body models. It is hence desirable to further expand the toolbox.  ...  The approach is independent from the choice of boundary conditions, and it does not assume a particular lattice structure. The probe is hence suitable for, e.g., fractals and quasicrystals.  ...  The Hofstadter model describes particles hopping on a two-dimensional square lattice in the presence of a magnetic field perpendicular to the plane.  ... 
arXiv:1909.02046v4 fatcat:ro6df2ln75c7zfbzfxgjqnsvbu

Perfect Sampling in Infinite Spin Systems via Strong Spatial Mixing [article]

Konrad Anand, Mark Jerrum
2021 arXiv   pre-print
The run-time is linear in the size of the window.  ...  The sampling algorithm assumes strong spatial mixing together with subexponential growth of G. It produces a finite window onto a perfect sample from the Gibbs distribution.  ...  by pioneering work on high dimensional expanders [KO20; ALGV18].  ... 
arXiv:2106.15992v1 fatcat:clwqvd22xveb5b7cwup7tntlui
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