Global and local scaling limits for the β = 2 Stieltjes–Wigert random matrix ensemble

2021 ◽  
pp. 2250020
Author(s):  
Peter J. Forrester
Keyword(s):  
Statistical Physics ◽  
Scaling Limit ◽  
Pair Potential ◽  
Exact Expression ◽  
Correlation Kernel ◽  
Local Scaling ◽  
Gaussian Unitary Ensemble ◽  
Matrix Ensemble ◽  
Many Body Systems ◽  
Global And Local

The eigenvalue probability density function (PDF) for the Gaussian unitary ensemble has a well-known analogy with the Boltzmann factor for a classical log-gas with pair potential [Formula: see text], confined by a one-body harmonic potential. A generalization is to replace the pair potential by [Formula: see text]. The resulting PDF first appeared in the statistical physics literature in relation to non-intersecting Brownian walkers, equally spaced at time [Formula: see text], and subsequently in the study of quantum many-body systems of the Calogero–Sutherland type, and also in Chern–Simons field theory. It is an example of a determinantal point process with correlation kernel based on the Stieltjes–Wigert polynomials. We take up the problem of determining the moments of this ensemble, and find an exact expression in terms of a particular little [Formula: see text]-Jacobi polynomial. From their large [Formula: see text] form, the global density can be computed. Previous work has evaluated the edge scaling limit of the correlation kernel in terms of the Ramanujan ([Formula: see text]-Airy) function. We show how in a particular [Formula: see text] scaling limit, this reduces to the Airy kernel.

scholarly journals THE CRITICAL POINT OF UNORIENTED RANDOM SURFACES WITH A NONEVEN POTENTIAL

1993 ◽  
Vol 08 (06) ◽  
pp. 1139-1152
Author(s):  
M.A. MARTÍN-DELGADO
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Critical Point ◽  
Discrete Model ◽  
Recursion Relation ◽  
Scaling Limit ◽  
Random Surfaces ◽  
Matrix Ensemble ◽  
Quartic Interaction ◽  
Double Scaling Limit

The discrete model of the real symmetric one-matrix ensemble is analyzed with a cubic interaction. The partition function is found to satisfy a recursion relation that solves the model. The double scaling-limit of the recursion relation leads to a Miura transformation relating the contributions to the free energy coming from oriented and unoriented random surfaces. This transformation is the same kind as found with a quartic interaction.

DISPERSIVE FRACTAL CHARACTERIZATION OF KIDNEY ARTERIES BY THREE-DIMENSIONAL MASS-RADIUS-ANALYSIS

Fractals ◽  
1995 ◽  
Vol 03 (04) ◽  
pp. 879-891 ◽  
Author(s):  
M. SERNETZ ◽  
M. JUSTEN ◽  
F. JESTCZEMSKI
Keyword(s):  
Fractal Structure ◽  
Three Dimensional ◽  
Arterial System ◽  
Data Sets ◽  
Scaling Properties ◽  
Local Scaling ◽  
Living Organisms ◽  
Global And Local ◽  
First Time

Three-dimensional data sets of kidney arterial vessels were obtained from resin casts by serial sectioning and by micro-NMR-tomography, and were analyzed by the mass-radius-relation both for global and local scaling properties. We present for the first time the spatial resolution of local scaling and thus the dispersion of the fractal dimension within the organs. The arterial system is characterized as a non-homogeneous fractal. We discuss and relate the fractal structure to the scaling and allometry of metabolic rates in living organisms.

Global and local scaling analysis of link streams in access and backbone core networks

2019 ◽  
Vol 149 ◽  
pp. 154-172 ◽  
Author(s):  
Asad Arfeen ◽  
Krzysztof Pawlikowski ◽  
Don McNickle ◽  
Andreas Willig
Keyword(s):  
Scaling Analysis ◽  
Local Scaling ◽  
Core Networks ◽  
Global And Local

scholarly journals Scattering in Quantum Dots via Noncommutative Rational Functions

2021 ◽  
Author(s):  
László Erdős ◽  
Torben Krüger ◽  
Yuriy Nemish
Keyword(s):  
Quantum Dots ◽  
Degrees Of Freedom ◽  
Joint Probability ◽  
Rational Functions ◽  
Borderline Case ◽  
Random Matrix Model ◽  
The Matrix ◽  
Matrix Ensemble ◽  
Rational Expressions ◽  
Global And Local

AbstractIn the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via $$N\ll M$$ N ≪ M channels, the density $$\rho $$ ρ of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large N regime allowing for (i) arbitrary ratio $$\phi := N/M\le 1$$ ϕ : = N / M ≤ 1 ; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit $$\phi \rightarrow 0$$ ϕ → 0 , we recover the formula for the density $$\rho $$ ρ that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker’s formula persists for any $$\phi <1$$ ϕ < 1 but in the borderline case $$\phi =1$$ ϕ = 1 an anomalous $$\lambda ^{-2/3}$$ λ - 2 / 3 singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries.

Combining global and local scaling (3D) methods to detect pore spaces in CT soil images

2021 ◽  
Author(s):  
Juan José Martin Sotoca ◽  
Antonio Saa Requejo ◽  
Sergio Zubelzu ◽  
Ana M. Tarquis
Keyword(s):  
Image Segmentation ◽  
Maximum Entropy ◽  
Maximum Entropy Method ◽  
Pore Space ◽  
Entropy Method ◽  
Local Scaling ◽  
Fractal Properties ◽  
Segmentation Methods ◽  
Soil Pore Space ◽  
Global And Local

&lt;p&gt;The characterization of the spatial distribution of soil pore structures is essential to obtain different parameters that will be useful in developing predictive models for a range of physical, chemical, and biological processes in soils. Over the last decade, major technological advances in X-ray computed tomography (CT) have allowed for the investigation and reconstruction of natural porous soils at very fine scales. Delimiting the pore structure (pore space) from the CT soil images applying image segmentation methods is crucial when attempting to extract complex pore space geometry information.&lt;/p&gt;&lt;p&gt;Different segmentation methods can result in different spatial distributions of pores influencing the parameters used in the models [1]. A new combined global &amp; local segmentation (2D) method called &amp;#8220;Combining Singularity-CA method&amp;#8221; was successfully applied [2]. This method combines a local scaling method (Singularity-CA method) with a global one (Maximum Entropy method). The Singularity-CA method, based on fractal concepts, creates singularity maps, and the CA (Concentration Area) method is used to define local thresholds that can be applied to binarize CT images [3]. Comparing Singularity-CA method with classical methods, such as Otsu and Maximum Entropy, we observed that more pores can be detected mainly due to its ability to amplify anomalous concentrations. However, some small pores were detected incorrectly. Combining Singularity-CA (2D) method gives better pore detection performance than the Singularity-CA and the Maximum Entropy method applied individually to the images.&lt;/p&gt;&lt;p&gt;The Combining Singularity-CV (3D) method is presented in this work. It combines the Singularity &amp;#8211; CV (Concentration Volume) method [4] and a global one to improve 3D pore space detection.&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;References:&lt;/p&gt;&lt;p&gt;[1] Zhang, Y.J. (2001). A review of recent evaluation methods for image segmentation: International symposium on signal processing and its applications. Kuala Lumpur, Malaysia, 13&amp;#8211;16, pp. 148&amp;#8211;151.&lt;/p&gt;&lt;p&gt;[2] Mart&amp;#237;n-Sotoca, J.J., Saa-Requejo, A., Grau, J.B., Paz-Gonz&amp;#225;lez, A., and Tarquis, A.M. (2018). Combining global and local scaling methods to detect soil pore space. J. of Geo. Exploration, vol. 189, June 2018, pp 72-84.&lt;/p&gt;&lt;p&gt;[3] Mart&amp;#237;n-Sotoca, J.J., Saa-Requejo, A., Grau, J.B. and Tarquis, A.M. (2017). New segmentation method based on fractal properties using singularity maps. Geoderma, vol. 287, February 2017, pp 40-53. http://dx.doi.org/10.1016/j.geoderma.2016.09.005.&lt;/p&gt;&lt;p&gt;[4] Mart&amp;#237;n-Sotoca, J.J., Saa-Requejo, A., Grau, J.B. and Tarquis, A.M. (2018). Local 3D segmentation of soil pore space based on fractal properties using singularity maps. Geoderma, vol. 311, February 2018, pp 175-188. http://dx.doi.org/10.1016/j.geoderma.2016.11.029.&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;Acknowledgements:&lt;/p&gt;&lt;p&gt;The authors acknowledge support from Project No. PGC2018-093854-B-I00 of the Spanish Ministerio de Ciencia Innovaci&amp;#243;n y Universidades of Spain and the funding from the Comunidad de Madrid (Spain), Structural Funds 2014-2020 512 (ERDF and ESF), through project AGRISOST-CM S2018/BAA-4330.&lt;/p&gt;

The neutral relativistic scalar field

2021 ◽  
pp. 105-124
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Field Theory ◽  
Real Time ◽  
Statistical Physics ◽  
Relativistic Quantum ◽  
Pair Potential ◽  
Field Extension ◽  
Bose Gas ◽  
Real Field ◽  
Relativistic Limit ◽  
Grand Canonical

This chapter introduces the relativistic quantum field theory (QFT) of the neutral scalar boson field. It is a local, relativistic invariant, theory for a real field extension of the non-relativistic field theory of the Bose gas. Locality is a property that plays a central role in most of this work. The QFT is discussed both from the viewpoint of real-time evolution and statistical physics. The holomorphic formalism leads to representations of the S-matrix in terms of field integrals. The S-matrix elements are related to the continuation to real time of various kinds of Euclidean correlation functions. It is argued that the massive φ4 QFT has the quantum Bose gas with a pair potential, in the grand canonical formulation, as a non-relativistic limit.

scholarly journals Product Matrix Processes as Limits of Random Plane Partitions

2019 ◽  
Vol 2020 (20) ◽  
pp. 6713-6768
Author(s):  
Alexei Borodin ◽  
Vadim Gorin ◽  
Eugene Strahov
Keyword(s):  
Scaling Limit ◽  
Singular Values ◽  
Continuous Limit ◽  
Dynamical Correlation ◽  
Unitary Matrices ◽  
Correlation Kernel ◽  
Plane Partitions ◽  
Product Matrix ◽  
Schur Process ◽  
Random Unitary Matrices

AbstractWe consider a random process with discrete time formed by squared singular values of products of truncations of Haar-distributed unitary matrices. We show that this process can be understood as a scaling limit of the Schur process, which gives determinantal formulas for (dynamical) correlation functions and a contour integral representation for the correlation kernel. The relation with the Schur processes implies that the continuous limit of marginals for q-distributed plane partitions coincides with the joint law of squared singular values for products of truncations of Haar-distributed random unitary matrices. We provide structural reasons for this coincidence that may also extend to other classes of random matrices.

Combining global and local scaling methods to detect soil pore space

2018 ◽  
Vol 189 ◽  
pp. 72-84 ◽  
Author(s):  
J.J. Martín-Sotoca ◽  
A. Saa-Requejo ◽  
J.B. Grau ◽  
A. Paz-González ◽  
A.M. Tarquis
Keyword(s):  
Pore Space ◽  
Local Scaling ◽  
Scaling Methods ◽  
Soil Pore Space ◽  
Global And Local
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