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Abstract: Abstract The preimage entropy provides a quantitative estimate of how “invertible” a system is. Once there are several examples where the preimage entropy is infinite, it cannot provide more information. Thus, we introduce the concept of preimage metric mean dimension, and study many properties of it. Meanwhile, we provide many examples to compute their preimage mean metric dimension. PubDate: 2024-02-25

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Abstract: Abstract We consider a chain consisting of \(n+1\) harmonic oscillators subjected on the right to a time dependent periodic force \({{\mathcal {F}}}(t)\) while Langevin thermostats are attached at both endpoints of the chain. We show that for long times the system is described by a Gaussian measure whose covariance function is independent of the force, while the means are periodic. We compute explicitly the work and energy due to the periodic force for all n including \(n\rightarrow \infty \) . PubDate: 2024-02-24

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Abstract: Abstract This paper develops approximate message passing algorithms to optimize multi-species spherical spin glasses. We first show how to efficiently achieve the algorithmic threshold energy identified in our companion work (Huang and Sellke in arXiv preprint, 2023. arXiv:2303.12172), thus confirming that the Lipschitz hardness result proved therein is tight. Next we give two generalized algorithms which produce multiple outputs and show all of them are approximate critical points. Namely, in an r-species model we construct \(2^r\) approximate critical points when the external field is stronger than a “topological trivialization" phase boundary, and exponentially many such points in the complementary regime. We also compute the local behavior of the Hamiltonian around each. These extensions are relevant for another companion work (Huang and Sellke in arXiv preprint, 2023. arXiv:2308.09677) on topological trivialization of the landscape. PubDate: 2024-02-24

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Abstract: Abstract We establish relations between different characterizations of order in spin glass models. We first prove that the broadening of the replica overlap distribution indicated by a nonzero standard deviation of the replica overlap \(R^{1,2}\) implies the non-differentiability of the two-replica free energy with respect to the replica coupling parameter \(\lambda \) . In \({\mathbb {Z}}_2\) invariant models such as the standard Edwards–Anderson model, the non-differentiability is equivalent to the spin glass order characterized by a nonzero Edwards–Anderson order parameter. This generalization of Griffiths’ theorem is proved for any short-range spin glass models with classical bounded spins. We also prove that the non-differentiability of the two-replica free energy mentioned above implies replica symmetry breaking in the literal sense, i.e., a spontaneous breakdown of the permutation symmetry in the model with three replicas. This is a general result that applies to a large class of random spin models, including long-range models such as the Sherrington-Kirkpatrick model and the random energy model. There is a 25-minute video that explains the main results of the present work: https://youtu.be/BF3hJiY1xvI PubDate: 2024-02-23

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Abstract: Abstract We study the stochastic dynamics of a two-dimensional particle whose coordinates are described by two coupled one-dimensional random-acceleration processes, that evolve in a confining parabolic potential and are subject to independent Gaussian white noises with different amplitudes (temperatures). We first determine standard characteristics: the mixed moments of positions and velocities, as well as the position-velocity probability density function (PDF) and those of its kinetic and potential energies. Going then beyond these standard characteristics, we consider the emerging rotational motion of the particle around the origin: We show that if the amplitudes of the noises are not equal, the particle experiences a non-zero (on average) torque, such that the angular momentum L and the angular velocity W have non-zero mean values which both are (irregularly) oscillating with time t. We evaluate the PDF-s of L and W and show that the former has exponential tails for any fixed t, and hence, all moments. In the large-time limit this PDF converges to a uniform distribution with a diverging variance. The PDF of W possesses heavy power-law tails such that the mean W is the only existing moment. However, this PDF converges to a well-defined large-time limit manifesting the possibility of stabilizing phenomenon even in frictionless driven systems. Surprisingly, the limit is independent of the amplitudes of noises. PubDate: 2024-02-23

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Abstract: Abstract We consider a family of models having an arbitrary positive amount of mass on each site and randomly exchanging an arbitrary amount of mass with nearest neighbor sites. We restrict to the case of diffusive models. We identify a class of reversible models for which the product invariant measure is known and the gradient condition is satisfied so that we can explicitly compute the transport coefficients associated to the diffusive hydrodynamic rescaling. Based on the Macroscopic Fluctuation Theory (Bertini et al. in Rev Mod Phys 87:593–636, 2015) we have that the large deviations rate functional for a stationary non equilibrium state can be computed solving a Hamilton–Jacobi equation depending only on the transport coefficients and the details of the boundary sources. Thus, we are able to identify a class of models having transport coefficients for which the Hamilton–Jacobi equation can indeed be solved. We give a complete characterization in the case of generalized zero range models and discuss several other cases. For the generalized zero range models we identify a class of discrete models that, modulo trivial extensions, coincides with the class discussed in Frassek and Giardinà (J Math Phys 63(10):103301–103335, 2022) and a class of continuous dynamics that coincides with the class in Franceschini et al. (J Math Phys 64(4): 043304–043321, 2023). Along the discussion we obtain a complete characterization of reversible misanthrope processes solving the discrete equations in Cocozza-Thivent (Z Wahrsch Verw Gebiete 70(4):509–523, 1985). PubDate: 2024-02-21

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Abstract: Abstract We consider \(N\times N\) self-adjoint Gaussian random matrices defined by an arbitrary deterministic sparsity pattern with d nonzero entries per row. We show that such random matrices exhibit a canonical localization–delocalization transition near the edge of the spectrum: when \(d\gg \log N\) the random matrix possesses a delocalized approximate top eigenvector, while when \(d\ll \log N\) any approximate top eigenvector is localized. The key feature of this phenomenon is that it is universal with respect to the sparsity pattern, in contrast to the delocalization properties of exact eigenvectors which are sensitive to the specific sparsity pattern of the random matrix. PubDate: 2024-02-21

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Abstract: Abstract In this paper, we study the central limit theorem for the solutions of stochastic differential delay equations with small noises. Our aim is to provide explicit estimates for the rate of convergence in total variation distance. We also show that the convergence rate is of optimal order. PubDate: 2024-02-21

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Abstract: Abstract We analytically investigate the zero-temperature hysteresis loops in a one-dimensional 3-state clock model in the low disorder limit, incorporating dilution and quenched spins. Utilizing a probabilistic approach with continuous random fields, we study the behavior of the model at nearly zero frequency limit of the applied field and confirming our analytical results via simulations. Dilution reproduces distorted hysteresis loop shapes akin to those in geological magnetic rocks, while quenched spins significantly contribute to hysteresis loop asymmetry. PubDate: 2024-02-20

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Abstract: Abstract This paper develops an averaging approach on macroscopic scales to derive Smoluchowski–Kramers approximation for a Langevin equation with state dependent friction in d-dimensional space. In this approach we couple the microscopic dynamics to the macroscopic scales. The weak convergence rate is also presented. PubDate: 2024-02-17

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Abstract: Abstract This paper deals with the problem of finding the Fokker Planck Equation (FPE) for the single-time probability density function (PDF) that optimally approximates the single-time PDF of a 1-D Stochastic Differential Equation (SDE) with Gaussian correlated noise. In this context, we tackle two main tasks. First, we consider the case of weak noise and in this framework we give a formal ground to the effective correction, introduced elsewhere (Bianucci and Mannella in J Phys Commun 4(10):105019, 2020, https://doi.org/10.1088/2399-6528/abc54e), to the Best Fokker Planck Equation (a standard “Born-Oppenheimer” result), also covering the more general cases of multiplicative SDE. Second, we consider the FPE obtained by using the Local Linearization Approach (LLA), and we show that a generalized cumulant approach allows an understanding of why the LLA FPE performs so well, even for noises with long (but finite) time scales and large intensities. PubDate: 2024-02-15

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Abstract: Abstract The dissipative spectral form factor (DSFF), recently introduced in Li et al. (Phys Rev Lett 127(17):170602, 2021) for the Ginibre ensemble, is a key tool to study universal properties of dissipative quantum systems. In this work we compute the DSFF for a large class of random matrices with real or complex entries up to an intermediate time scale, confirming the predictions from Li et al. (Phys Rev Lett 127(17):170602, 2021). The analytic formula for the DSFF in the real case was previously unknown. Furthermore, we show that for short times the connected component of the DSFF exhibits a non-universal correction depending on the fourth cumulant of the entries. These results are based on the central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices Cipolloni et al. (Electron J Prob 26:1–61, 2021) and Cipolloni et al. (Commun Pure Appl Math 76(5): 946–1034, 2023). PubDate: 2024-02-15

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Abstract: Abstract We study the radius of gyration \(R_T\) of a self-repellent fractional Brownian motion \(\left\{ B^H_t\right\} _{0\le t\le T}\) taking values in \(\mathbb {R}^d\) . Our sharpest result is for \(d=1\) , where we find that with high probability, $$\begin{aligned} R_T \asymp T^\nu , \quad \text {with }\quad \nu =\frac{2}{3}\left( 1+H\right) . \end{aligned}$$ For \(d>1\) , we provide upper and lower bounds for the exponent \(\nu \) , but these bounds do not match. PubDate: 2024-01-31

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Abstract: Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode non-dissipative effects. Our main contribution is an abstract theory, which for a given flux-density cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems—independent copies of jump processes, zero-range processes, chemical-reaction networks in complex balance and lattice-gas models—without assuming detailed balance. For macroscopic equations arising out of these particle systems, we derive new variational formulations that generalise the classical gradient-flow formulation. PubDate: 2024-01-29

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Abstract: Abstract We develop a novel cluster expansion for finite-spin lattice systems subject to multi-body quantum —and, in particular, classical— interactions. Our approach is based on the use of “decoupling parameters”, advocated by Park (J. Stat. Phys. 27, 553–576 (1982)), which relates partition functions with successive additional interaction terms. Our treatment, however, leads to an explicit expansion in a \(\beta \) -dependent effective fugacity that permits an explicit evaluation of free energy and correlation functions at small \(\beta \) . To determine its convergence region we adopt a relatively recent cluster summation scheme that replaces the traditional use of Kikwood-Salzburg-like integral equations by more precise sums in terms of particular tree-diagrams Bissacot et al. (J. Stat. Phys. 139, 598–617 (2010)). As an application we show that our lower bound of the radius of \(\beta \) -analyticity is larger than Park’s for quantum systems two-body interactions. PubDate: 2024-01-28

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Abstract: Abstract We study the long time behaviour of a Brownian particle evolving in a dynamic random environment. Recently, Cannizzaro et al. (Ann Probab 50(6):2475–2498, 2022) proved sharp \(\sqrt{\log }\) -super diffusive bounds for a Brownian particle in the curl of (a regularisation of) the 2-D Gaussian Free Field (GFF) \(\underline{\omega }\) . We consider a one parameter family of Markovian and Gaussian dynamic environments which are reversible with respect to the law of \(\underline{\omega }\) . Adapting their method, we show that if \(s\ge 1\) , with \(s=1\) corresponding to the standard stochastic heat equation, then the particle stays \(\sqrt{\log }\) -super diffusive, whereas if \(s<1\) , corresponding to a fractional heat equation, then the particle becomes diffusive. In fact, for \(s<1\) , we show that this is a particular case of Komorowski and Olla (J Funct Anal 197(1):179–211, 2003), which yields an invariance principle through a Sector Condition result. Our main results agree with the Alder–Wainwright scaling argument (see Alder and Wainwright in Phys Rev Lett 18:988–990, 1967; Alder and Wainwright in Phys Rev A 1:18–21, 1970; Alder et al. in Phys Rev A 4:233–237, 1971; Forster et al. in Phys Rev A 16:732–749, 1977) used originally in Tóth and Valkó (J Stat Phys 147(1):113–131, 2012) to predict the \(\log \) -corrections to diffusivity. We also provide examples which display \(\log ^a\) -super diffusive behaviour for \(a\in (0,1/2]\) . PubDate: 2024-01-28

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Abstract: Abstract This paper is concerned with the stochastic reaction-diffusion lattice systems defined on the entire set with both locally Lipschitz nonlinear drift and diffusion terms. The central limit theorem is derived for such infinite-dimensional stochastic systems. Moreover, the moderate deviation principle of solution processes is also established by the weak convergence method based on the variational representation of positive functionals of Brownian motion. The method relies on proving the convergence of the solutions of the controlled stochastic lattice systems. PubDate: 2024-01-28

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Abstract: Abstract In traffic flow, flyovers play a vital role in reducing traffic congestion, advancing commute timing, and helping to prevent collision scenarios. Inspired by the real-life applications of flyover, we model a coupled two-lane transport system where the lanes are divided into three segments in which the particle lane switching mechanism executes only in the middle segment of the lane, and particle inclusion and expulsion kinetics are performed in first and last segment of the lane to embody the infrastructure of flyover. The nature of the system characteristics is analyzed for diverse values of particle inclusion, expulsion, and coupling rates through graphical diagrams of phase planes, density profiles, shock dynamics, and phase transitions. The time-invariant behavior of the system is inspected numerically by utilizing the well-known method of mean-field theory, and it is discovered that the biased dynamics of particle inclusion, expulsion, and fully asymmetric coupling condition yield the phase diagram to disclose more unanticipated mixed stationary phases. Besides, the system experiences the fruitful novel incident of the double shock phase when the possibility of particle inclusion occurs relatively higher than the expulsion event. Also, it is noticed that the system experiences the fascinating phenomenon of shock propagation by traveling in either vertical/horizontal direction in the shock region of the phase diagram under the symmetric coupling circumstance, such as the position of the shock suddenly transits from the right to left segment without crossing the middle segment. The calculated numerical results are in good agreement with Monte Carlo simulation. PubDate: 2024-01-28

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Abstract: Abstract We present a method to compute transport coefficients in molecular dynamics. Transport coefficients quantify the linear dependencies of fluxes in non-equilibrium systems subject to small external forcings. Whereas standard non-equilibrium approaches fix the forcing and measure the average flux induced in the system driven out of equilibrium, a dual philosophy consists in fixing the value of the flux, and measuring the average magnitude of the forcing needed to induce it. A deterministic version of this approach, named Norton dynamics, was studied in the 1980s by Evans and Morriss. In this work, we introduce a stochastic version of this method, first developing a general formal theory for a broad class of diffusion processes, and then specializing it to underdamped Langevin dynamics, which are commonly used for molecular dynamics simulations. We provide numerical evidence that the stochastic Norton method provides an equivalent measure of the linear response, and in fact demonstrate that this equivalence extends well beyond the linear response regime. This work raises many intriguing questions, both from the theoretical and the numerical perspectives. PubDate: 2024-01-28

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Abstract: Abstract It is rigorously shown that an appropriate quantum annealing for any finite-dimensional spin system has no quantum first-order transition in transverse magnetization. This result can be applied to finite-dimensional spin-glass systems, where the ground state search problem is known to be hard to solve. Consequently, it is strongly suggested that the quantum first-order transition in transverse magnetization is not fatal to the difficulty of combinatorial optimization problems in quantum annealing. PubDate: 2024-01-28