Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Second Derivatives and Laplace Operator01:22

Second Derivatives and Laplace Operator

2.6K
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
2.6K
Chebyshev's Theorem to Interpret Standard Deviation01:15

Chebyshev's Theorem to Interpret Standard Deviation

5.0K
Chebyshev’s theorem, also known as Chebyshev’s Inequality, states that the proportion of values of a dataset for K standard deviation is calculated using the equation:
5.0K
Routh-Hurwitz Criterion II01:19

Routh-Hurwitz Criterion II

906
In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
906
Gauss's Law: Cylindrical Symmetry01:20

Gauss's Law: Cylindrical Symmetry

9.2K
A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
9.2K
Euler's Formula for Pin-Ended Columns01:21

Euler's Formula for Pin-Ended Columns

658
In structural engineering, the stability of columns under compressive axial loads is a critical consideration, described as buckling. A typical example involves a column PQ, which is pin-connected at both ends and subjected to a centric axial load F applied at one end, with a reaction force of F' = -F at the other end. Here, it is crucial to understand that when an applied load exceeds the critical load, buckling occurs as the system becomes unstable.
To calculate the critical load, envision...
658
Residuals and Least-Squares Property01:11

Residuals and Least-Squares Property

8.9K
The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative, and the line overestimates the actual data value for y.
The process of fitting the best-fit...
8.9K

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Cusp Universality for Correlated Random Matrices.

Communications in mathematical physics·2025
Same author

Eigenstate Thermalization Hypothesis for Wigner-Type Matrices.

Communications in mathematical physics·2024
Same journal

Dissipative particle systems on expanders.

Probability theory and related fields·2026
Same journal

Mixing of fast random walks on dynamic random permutations.

Probability theory and related fields·2026
Same journal

An invariance principle for the 2<i>d</i> weakly self-repelling Brownian polymer.

Probability theory and related fields·2026
Same journal

Subexponential lower bounds for <i>f</i>-ergodic Markov processes.

Probability theory and related fields·2026
Same journal

Phase transition for random walks on graphs with added weighted random matching.

Probability theory and related fields·2025
Same journal

The Allen-Cahn equation with weakly critical random initial datum.

Probability theory and related fields·2025
See all related articles

Related Experiment Video

Updated: Jan 9, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
06:44

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

480

Linear Eigenvalue Statistics at the cusp.

Volodymyr Riabov1

  • 1IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria.

Probability Theory and Related Fields
|December 8, 2025
PubMed
Summary
This summary is machine-generated.

This study reveals universal Gaussian fluctuations in random matrix eigenvalue statistics near spectral density singularities. It provides a complete description of these statistics across all regimes, including cusps and regular edges.

Keywords:
Central limit theoremCuspEdgeMesoscopic eigenvalue statisticsWigner-type matrix

More Related Videos

Experimental and Data Analysis Workflow for Soft Matter Nanoindentation
13:04

Experimental and Data Analysis Workflow for Soft Matter Nanoindentation

Published on: January 18, 2022

4.7K
Author Spotlight: Advancing Hepatic Fibrosis Diagnosis Using Magnetic Resonance Elastography and AI
06:09

Author Spotlight: Advancing Hepatic Fibrosis Diagnosis Using Magnetic Resonance Elastography and AI

Published on: July 21, 2023

1.8K

Related Experiment Videos

Last Updated: Jan 9, 2026

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
06:44

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

480
Experimental and Data Analysis Workflow for Soft Matter Nanoindentation
13:04

Experimental and Data Analysis Workflow for Soft Matter Nanoindentation

Published on: January 18, 2022

4.7K
Author Spotlight: Advancing Hepatic Fibrosis Diagnosis Using Magnetic Resonance Elastography and AI
06:09

Author Spotlight: Advancing Hepatic Fibrosis Diagnosis Using Magnetic Resonance Elastography and AI

Published on: July 21, 2023

1.8K

Area of Science:

  • Mathematics
  • Physics
  • Statistical Mechanics

Background:

  • Wigner-type random matrices are fundamental in statistical mechanics and quantum chaos.
  • Spectral density singularities, particularly cusps, pose challenges for understanding eigenvalue statistics.
  • Previous studies lacked analysis of linear eigenvalue statistics at cusp-like singularities.

Purpose of the Study:

  • To establish universal Gaussian fluctuations for mesoscopic linear eigenvalue statistics near cusp-like singularities.
  • To analyze the transitionary regime from regular edges to sharp cusps and the bulk.
  • To provide a complete description of eigenvalue statistics in all possible regimes.

Main Methods:

  • Analysis of mesoscopic linear eigenvalue statistics.
  • Investigation of Wigner-type random matrices.
  • Development of new functionals for bias and variance.

Main Results:

  • Universal Gaussian fluctuations are established for eigenvalue statistics near cusps.
  • A new one-parameter family of functionals governing bias and variance is identified.
  • The analysis covers the entire transitionary regime, from regular edges to sharp cusps and the bulk.

Conclusions:

  • This work provides a complete description of linear eigenvalue statistics in all regimes of Wigner-type random matrices.
  • The identified functionals interpolate between known formulas for bulk and edge statistics.
  • The findings offer new insights into the behavior of random matrices near spectral singularities.