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Related Experiment Video

Updated: Nov 27, 2025

Improving the Success Rate of Protein Crystallization by Random Microseed Matrix Screening

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On Products of Random Matrices.

Natalia Amburg^1,2,3, Aleksander Orlov⁴, Dmitry Vasiliev^1,2,5

¹A.I. Alikhanov Institute for Theoretical and Experimental Physics of NRC Kurchatov Institute, B. Cheremushkinskaya, 25, 117259 Moscow, Russia.

Entropy (Basel, Switzerland)

|December 8, 2020

Summary

This study introduces matrix models based on dessins d'enfant (children's drawings) by attaching random matrices to graph edges. These models connect graph theory, random matrices, and combinatorics, offering new insights into mathematical structures.

Keywords:

Hurwitz number Schur polynomial generalized hypergeometric functions integrable systems matrix models products of random matrices random complex and random unitary matrices

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Area of Science:

Combinatorics
Graph Theory
Mathematical Physics

Background:

Dessins d'enfant are special graphs drawn on surfaces.
These graphs have vertices called stars and edges.

Purpose of the Study:

Introduce a novel family of matrix models.
Associate these models with dessins d'enfant.
Explore connections between graph theory and random matrix theory.

Main Methods:

Constructing multimatrix models by attaching random matrices to graph edges.
Incorporating source matrices at vertices as coupling constants.
Analyzing integrals expressed via the 'spectrum of stars'.

Main Results:

Developed matrix models based on dessins d'enfant.
Results involve the 'spectrum of stars' and combinatorial numbers.
Potential inclusion of Hurwitz numbers and group representation characters.

Conclusions:

The proposed models offer a new framework linking dessins d'enfant with random matrix theory.
The 'spectrum of stars' provides a key quantity for understanding these models.
The models have implications for combinatorics and mathematical physics.