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1. Out-of-time-order correlators and Lyapunov exponents in sparse SYK

   https://doi.org/10.1007/JHEP11(2023)088  

   Cáceres, Elena; Guglielmo, Tyler; Kent, Brian; Misobuchi, Anderson (November 2023, Journal of High Energy Physics)

   A<sc>bstract</sc> We use a combination of analytical and numerical methods to study out-of-time order correlators (OTOCs) in the sparse Sachdev-Ye-Kitaev (SYK) model. We find that at a given order ofN, the standard result for theq-local, all-to-all SYK, obtained through the sum over ladder diagrams, is corrected by a series in the sparsity parameter,k. We present an algorithm to sum the diagrams at any given order of 1/(kq)ⁿ. We also study OTOCs numerically as a function of the sparsity parameter and determine the Lyapunov exponent. We find that numerical stability when extracting the Lyapunov exponent requires averaging over a massive number of realizations. This trade-off between the efficiency of the sparse model and consistent behavior at finiteNbecomes more significant for larger values ofN. 

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2. Quantum information scrambling and chemical reactions

   https://doi.org/10.1073/pnas.2321668121  

   Zhang, Chenghao; Kundu, Sohang; Makri, Nancy; Gruebele, Martin; Wolynes, Peter G (April 2024, Proceedings of the National Academy of Sciences)

   The ultimate regularity of quantum mechanics creates a tension with the assumption of classical chaos used in many of our pictures of chemical reaction dynamics. Out-of-time-order correlators (OTOCs) provide a quantum analog to the Lyapunov exponents that characterize classical chaotic motion. Maldacena, Shenker, and Stanford have suggested a fundamental quantum bound for the rate of information scrambling, which resembles a limit suggested by Herzfeld for chemical reaction rates. Here, we use OTOCs to study model reactions based on a double-well reaction coordinate coupled to anharmonic oscillators or to a continuum oscillator bath. Upon cooling, as one enters the tunneling regime where the reaction rate does not strongly depend on temperature, the quantum Lyapunov exponent can approach the scrambling bound and the effective reaction rate obtained from a population correlation function can approach the Herzfeld limit on reaction rates: Tunneling increases scrambling by expanding the state space available to the system. The coupling of a dissipative continuum bath to the reaction coordinate reduces the scrambling rate obtained from the early-time OTOC, thus making the scrambling bound harder to reach, in the same way that friction is known to lower the temperature at which thermally activated barrier crossing goes over to the low-temperature activationless tunneling regime. Thus, chemical reactions entering the tunneling regime can be information scramblers as powerful as the black holes to which the quantum Lyapunov exponent bound has usually been applied. 

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3. Sparse SYK and traversable wormholes

   https://doi.org/10.1007/JHEP11(2021)015  

   Cáceres, Elena; Misobuchi, Anderson; Pimentel, Rafael (November 2021, Journal of High Energy Physics)

   A bstract We investigate two sparse Sachdev-Ye-Kitaev (SYK) systems coupled by a bilinear term as a holographic quantum mechanical description of an eternal traversable wormhole in the low temperature limit. Each SYK system consists of N Majorana fermions coupled by random q -body interactions. The degree of sparseness is captured by a regular hypergraph in such a way that the Hamiltonian contains exactly k N independent terms. We improve on the theoretical understanding of the sparseness property by using known measures of hypergraph expansion. We show that the sparse version of the two coupled SYK model is gapped with a ground state close to a thermofield double state. Using Krylov subspace and parallelization techniques, we simulate the system for q = 4 and q = 8. The sparsity of the model allows us to explore larger values of N than the ones existing in the literature for the all-to-all SYK. We analyze in detail the two-point functions and the transmission amplitude of signals between the two systems. We identify a range of parameters where revivals obey the scaling predicted by holography and signals can be interpreted as traversing the wormhole. 

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4. Quantum chaos in a weakly-coupled field theory with nonlocality

   https://doi.org/10.1007/JHEP09(2022)097  

   Fischler, Willy; Guglielmo, Tyler; Nguyen, Phuc (September 2022, Journal of High Energy Physics)

   A bstract In order to study the chaotic behavior of a system with non-local interactions, we will consider weakly coupled non-commutative field theories. We compute the Lyapunov exponent of this exponential growth in the large Moyal-scale limit to leading order in the t’Hooft coupling and 1/ N . We found that in this limit, the Lyapunov exponent remains comparable in magnitude to (and somewhat smaller than) the exponent in the commutative case. This can possibly be explained by the infrared sensitivity of the Lyapunov exponent. Another possible explanation is that in examples of weakly coupled non-commutative field theories, non-local contributions to various thermodynamic quantities are sub-dominant. 

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5. Sparse confidence sets for normal mean models

   https://doi.org/10.1093/imaiai/iaad003  

   Ning, Yang; Cheng, Guang (March 2023, Information and Inference: A Journal of the IMA)

   Abstract In this paper, we propose a new framework to construct confidence sets for a $$d$$-dimensional unknown sparse parameter $${\boldsymbol \theta }$$ under the normal mean model $${\boldsymbol X}\sim N({\boldsymbol \theta },\sigma ^{2}\bf{I})$$. A key feature of the proposed confidence set is its capability to account for the sparsity of $${\boldsymbol \theta }$$, thus named as sparse confidence set. This is in sharp contrast with the classical methods, such as the Bonferroni confidence intervals and other resampling-based procedures, where the sparsity of $${\boldsymbol \theta }$$ is often ignored. Specifically, we require the desired sparse confidence set to satisfy the following two conditions: (i) uniformly over the parameter space, the coverage probability for $${\boldsymbol \theta }$$ is above a pre-specified level; (ii) there exists a random subset $$S$$ of $$\{1,...,d\}$$ such that $$S$$ guarantees the pre-specified true negative rate for detecting non-zero $$\theta _{j}$$’s. To exploit the sparsity of $${\boldsymbol \theta }$$, we allow the confidence interval for $$\theta _{j}$$ to degenerate to a single point 0 for any $$j\notin S$$. Under this new framework, we first consider whether there exist sparse confidence sets that satisfy the above two conditions. To address this question, we establish a non-asymptotic minimax lower bound for the non-coverage probability over a suitable class of sparse confidence sets. The lower bound deciphers the role of sparsity and minimum signal-to-noise ratio (SNR) in the construction of sparse confidence sets. Furthermore, under suitable conditions on the SNR, a two-stage procedure is proposed to construct a sparse confidence set. To evaluate the optimality, the proposed sparse confidence set is shown to attain a minimax lower bound of some properly defined risk function up to a constant factor. Finally, we develop an adaptive procedure to the unknown sparsity. Numerical studies are conducted to verify the theoretical results. 

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