Matrix models and growth processes: from viscous flows to the quantum Hall effect

A.Zabrodin

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Matrix models and growth processes: from viscous flows to the quantum Hall effect

A.Zabrodin

High Energy Physics - Theory · Submitted 19 Dec 2004

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim As far as the physical significance of these models is concerned, we discuss, in some detail, the recently revealed applications to diffusion-controlled growth processes (e.g., to the Saffman-Taylor problem) and to the semiclassical behaviour of electronic blobs in the quantum Hall regime.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.07 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.08 · mentions code or data availability 0.01.

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We review the recent developments in the theory of normal, normal self-dual and general complex random matrices. The distribution and correlations of the eigenvalues at large scales are investigated in the large $N$ limit. The 1/N expansion of the free energy is also discussed. Our basic tool is a specific Ward identity for correlation functions (the loop equation), which follows from invariance of the partition function under reparametrizations of the complex eigenvalues plane. The method for handling the loop equation requires the technique of boundary value problems in two dimensions and elements of the potential theory. As far as the physical significance of these models is concerned, we discuss, in some detail, the recently revealed applications to diffusion-controlled growth processes (e.g., to the Saffman-Taylor problem) and to the semiclassical behaviour of electronic blobs in the quantum Hall regime.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We review the recent developments in the theory of normal, normal self-dual and general complex random matrices.	contribution 0.68	0.19	0.05	asserted 0.73
s1	The distribution and correlations of the eigenvalues at large scales are investigated in the large $N$ limit.	contribution 0.94	0.62	0.06	asserted 0.96
s2	The 1/N expansion of the free energy is also discussed.	contribution 0.87	0.50	0.04	asserted 0.95
s3	Our basic tool is a specific Ward identity for correlation functions (the loop equation), which follows from invariance of the partition function under reparametrizations of the complex eigenvalues plane.	method 0.99	0.56	0.07	asserted 1.00
s4	The method for handling the loop equation requires the technique of boundary value problems in two dimensions and elements of the potential theory.	method 1.00	0.47	0.03	asserted 1.00
s5	As far as the physical significance of these models is concerned, we discuss, in some detail, the recently revealed applications to diffusion-controlled growth processes (e.g., to the Saffman-Taylor problem) and to the semiclassical behaviour of electronic blobs in the quantum Hall regime.	interpretation 0.80	0.28	0.24	asserted 0.96

What kind of paper the classifier reports this is: review 1.00 · evidence from proof or derivation 0.56.

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Original abstract

We review the recent developments in the theory of normal, normal self-dual and general complex random matrices. The distribution and correlations of the eigenvalues at large scales are investigated in the large $N$ limit. The 1/N expansion of the free energy is also discussed. Our basic tool is a specific Ward identity for correlation functions (the loop equation), which follows from invariance of the partition function under reparametrizations of the complex eigenvalues plane. The method for handling the loop equation requires the technique of boundary value problems in two dimensions and elements of the potential theory. As far as the physical significance of these models is concerned, we discuss, in some detail, the recently revealed applications to diffusion-controlled growth processes (e.g., to the Saffman-Taylor problem) and to the semiclassical behaviour of electronic blobs in the quantum Hall regime.

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References

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

(Un)solvable Matrix Models for BPS Correlators
hep-th 2025-08 unverdicted novelty 6.0 of 10

Proposes complex matrix models for BPS correlators in N=4 SYM, relating eigenvalue distributions to LLM droplet shapes and enabling computations of one-point functions and three-point correlators via reductions to kno...
Critical Lin-Lunin-Maldacena geometries
hep-th 2026-06 unverdicted novelty 5.0 of 10

Near a cusp in the LLM droplet, the geometry acquires a universal ISO(1,3)×SO(5) symmetric form with a naked singularity that traps both massless and massive particles, admitting analytic massless trajectories and hin...

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11 arXiv papers cite this in 19 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 15 sentences without a confident label

Cites it as background 3 papers

First we review two-point functions for characters and exponential operators which are well studied in the literature [ref] .

(Un)solvable Matrix Models for BPS Correlators · Two-Point Functions of Huge $1/2$-BPS operators · 1.00
In particular, they are related to the problem of laplacian growth and viscous flows [ref] , conformal mappings in two dimensions and integrable hierarchies [ref] and the Bergman kernel theory [ref] .

Random normal matrices, Bergman kernel and projective embeddings · Introduction · 0.83
For more details, we refer the reader to [ref] .

Bulk asymptotics for polyanalytic correlation kernels · Bergman kernels for analytic polynomials · 0.99

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