scholarly journals Inflation versus projection sets in aperiodic systems: the role of the window in averaging and diffraction

2020 ◽  
Vol 76 (5) ◽  
pp. 559-570
Author(s):  
Michael Baake ◽  
Uwe Grimm
Keyword(s):  
Model Systems ◽  
Internal Space ◽  
Infinite Volume ◽  
Project Method ◽  
Topical Review ◽  
Volume Limit ◽  
Recent Developments ◽  
Infinite Volume Limit ◽  
Quantities Of Interest

Tilings based on the cut-and-project method are key model systems for the description of aperiodic solids. Typically, quantities of interest in crystallography involve averaging over large patches, and are well defined only in the infinite-volume limit. In particular, this is the case for autocorrelation and diffraction measures. For cut-and-project systems, the averaging can conveniently be transferred to internal space, which means dealing with the corresponding windows. In this topical review, this is illustrated by the example of averaged shelling numbers for the Fibonacci tiling, and the standard approach to the diffraction for this example is recapitulated. Further, recent developments are discussed for cut-and-project structures with an inflation symmetry, which are based on an internal counterpart of the renormalization cocycle. Finally, a brief review is given of the notion of hyperuniformity, which has recently gained popularity, and its application to aperiodic structures.

scholarly journals Tree Polymers in the Infinite Volume Limit at Critical Strong Disorder

2011 ◽  
Vol 48 (03) ◽  
pp. 885-891
Author(s):  
Torrey Johnson ◽  
Edward C. Waymire
Keyword(s):  
Asymptotic Variance ◽  
Standard Theory ◽  
Convergence In Probability ◽  
Infinite Volume ◽  
Weak Disorder ◽  
Branching Random Walks ◽  
Strong Disorder ◽  
Volume Limit ◽  
Specific Formula ◽  
Infinite Volume Limit

The almost-sure existence of a polymer probability in the infinite volume limit is readily obtained under general conditions of weak disorder from standard theory on multiplicative cascades or branching random walks. However, speculations in the case of strong disorder have been mixed. In this note existence of an infinite volume probability is established at critical strong disorder for which one has convergence in probability. Some calculations in support of a specific formula for the almost-sure asymptotic variance of the polymer path under strong disorder are also provided.

scholarly journals Non-integrable Ising Models in Cylindrical Geometry: Grassmann Representation and Infinite Volume Limit

2021 ◽  
Author(s):  
Giovanni Antinucci ◽  
Alessandro Giuliani ◽  
Rafael L. Greenblatt
Keyword(s):  
Nearest Neighbor ◽  
Scaling Limit ◽  
Ising Models ◽  
Infinite Volume ◽  
Effective Potentials ◽  
Volume Limit ◽  
Cylindrical Domains ◽  
Key Aspects ◽  
Infinite Volume Limit ◽  
Detailed Derivation

AbstractIn this paper, meant as a companion to Antinucci et al. (Energy correlations of non-integrable Ising models: the scaling limit in the cylinder, 2020. arXiv: 1701.05356), we consider a class of non-integrable 2D Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green’s function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries.

scholarly journals QUANTUM NOISE INDUCED ENTANGLEMENT AND CHAOS IN THE DISSIPATIVE QUANTUM MODEL OF BRAIN

2004 ◽  
Vol 18 (06) ◽  
pp. 841-858 ◽  
Author(s):  
ELIANO PESSA ◽  
GIUSEPPE VITIELLO
Keyword(s):  
Degrees Of Freedom ◽  
Quantum Noise ◽  
Random Force ◽  
Quantum Model ◽  
Infinite Volume ◽  
Quantum Dissipation ◽  
Memory Space ◽  
Volume Limit ◽  
System Ground State ◽  
Infinite Volume Limit

We discuss some features of the dissipative quantum model of brain in the frame of the formalism of quantum dissipation. Such a formalism is based on the doubling of the system degrees of freedom. We show that the doubled modes account for the quantum noise in the fluctuating random force in the system-environment coupling. Remarkably, such a noise manifests itself through the coherent structure of the system ground state. The entanglement of the system modes with the doubled modes is shown to be permanent in the infinite volume limit. In such a limit the trajectories in the memory space are classical chaotic trajectories.

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