scholarly journals Random matrices and Laplacian growth

2018 ◽  
Author(s):  
Patrik L. Ferrari ◽  
Herbert Spohn
Keyword(s):  
Random Matrices ◽  
Blow Up ◽  
Two Dimensions ◽  
Laplacian Growth ◽  
Normal Matrices ◽  
Large N ◽  
Complex Eigenvalues ◽  
General Complex ◽  
Exact Relations ◽  
Beta Ensembles

This article reviews the theory of random matrices with eigenvalues distributed in the complex plane and more general ‘beta ensembles’ (logarithmic gases in 2D). It first considers two ensembles of random matrices with complex eigenvalues: ensemble C of general complex matrices and ensemble N of normal matrices. In particular, it describes the Dyson gas picture for ensembles of matrices with general complex eigenvalues distributed on the plane. It then presents some general exact relations for correlation functions valid for any values of N and β before analysing the distribution and correlations of the eigenvalues in the large N limit. Using the technique of boundary value problems in two dimensions and elements of the potential theory, the article demonstrates that the finite-time blow-up (a cusp–like singularity) of the Laplacian growth with zero surface tension is a critical point of the normal and complex matrix models.

scholarly journals Large-N quantum gauge theories in two dimensions

1993 ◽  
Vol 303 (1-2) ◽  
pp. 95-98 ◽  
Author(s):  
B. Rusakov
Keyword(s):  
Gauge Theories ◽  
Two Dimensions ◽  
Large N

scholarly journals Random matrices in 2D, Laplacian growth and operator theory

2008 ◽  
Vol 41 (26) ◽  
pp. 263001 ◽  
Author(s):  
Mark Mineev-Weinstein ◽  
Mihai Putinar ◽  
Razvan Teodorescu
Keyword(s):  
Random Matrices ◽  
Operator Theory ◽  
Laplacian Growth

Mass generation in the large N Gross-Neveu model: a constructive proof without intermediate field

2021 ◽  
Author(s):  
Jeremie M. Unterberger
Keyword(s):  
Discrete Symmetry ◽  
Two Dimensions ◽  
Constructive Proof ◽  
Point Function ◽  
Large Component ◽  
Mass Generation ◽  
Bosonic Field ◽  
Large N ◽  
Discrete Symmetry Breaking ◽  
Gross Neveu Model

Abstract We give a new constructive proof of the infrared behavior of the Euclidean Gross-Neveu model in two dimensions with small coupling and large component number N. Our argument does not rely on the use of an intermediate (auxiliary bosonic) field. Instead bubble series are resummed by hand, and determinant bounds replaced by a control of local factorials relying on combinatorial arguments and Pauli's principle. The discrete symmetry-breaking is ensured by considering the model directly with a mass counterterm chosen in such a way as to cancel tadpole diagrams. Then the fermion two-point function is shown to decay (quasi-)exponentially as in [12]/

scholarly journals STRING HAMILTONIAN FROM GENERALIZED YANG–MILLS GAUGE THEORY IN TWO DIMENSIONS

2004 ◽  
Vol 19 (02) ◽  
pp. 205-225 ◽  
Author(s):  
FLORIAN DUBATH ◽  
SIMONE LELLI ◽  
ANNA RISSONE
Keyword(s):  
Gauge Theory ◽  
String Theory ◽  
Space Time ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Bosonic Field ◽  
Standard Case ◽  
Free Fermions ◽  
Yang Mills ◽  
Large N

Two-dimensional SU (N) Yang–Mills theory is known to be equivalent to a string theory, as found by Gross in the large N limit, using the 1/N expansion. Later it was found that even a generalized YM theory leads to a string theory of the Gross type. In the standard YM theory case, Douglas and others found the string Hamiltonian describing the propagation and the interactions of states made of strings winding on a cylindrical space–time. We address the problem of finding a similar Hamiltonian for the generalized YM theory. As in the standard case we start by writing the theory as a theory of free fermions. Performing a bosonization, we express the Hamiltonian in terms of the modes of a bosonic field, that are interpreted as in the standard case as creation and destruction operators for states of strings winding around the cylindrical space–time. The result is similar to the standard Hamiltonian, but with new kinds of interaction vertices.

Blow up problem for Landau-Lifshitz equations in two dimensions

2000 ◽  
Vol 5 (1) ◽  
pp. 43-44 ◽  
Author(s):  
Boling Guo ◽  
Yongqian Han ◽  
Ganshan Yang
Keyword(s):  
Blow Up ◽  
Two Dimensions

BLOW-UP SOLUTIONS OF THE TWO-DIMENSIONAL HEAT EQUATION DUE TO A LOCALIZED MOVING SOURCE

2005 ◽  
Vol 03 (01) ◽  
pp. 1-16 ◽  
Author(s):  
C. M. KIRK ◽  
W. E. OLMSTEAD
Keyword(s):  
Heat Equation ◽  
Source Term ◽  
Flat Surface ◽  
Blow Up ◽  
Heat Treating ◽  
Two Dimensions ◽  
Combustible Material ◽  
Localized Source ◽  
Diffusive Medium ◽  
The Mathematical Model

The problem examined is that of a localized energy source which undergoes planar motion along the surface of a reactive-diffusive medium. This is representative of a laser beam that is moving across the flat surface of a combustible material during a cutting, welding or heat treating process. The mathematical model for this situation is a heat equation in two-dimensions with a nonlinear source term, which is localized around a reference point that is allowed to move. Results are derived that indicate the roles played by the size, strength and motion of the localized source in determining whether or not a blow-up occurs.

scholarly journals Integrability-preserving regularizations of Laplacian Growth

2020 ◽  
Vol 15 ◽  
pp. 9
Author(s):  
Razvan Teodorescu
Keyword(s):  
Universality Class ◽  
Two Dimensions ◽  
Laplacian Growth ◽  
Boundary Singularity ◽  
Diffusion Limited Aggregation ◽  
Scaling Properties ◽  
Scale Free ◽  
Diffusion Limited ◽  
Long Time ◽  
Discrete Counterpart

The Laplacian Growth (LG) model is known as a universality class of scale-free aggregation models in two dimensions, characterized by classical integrability and featuring finite-time boundary singularity formation. A discrete counterpart, Diffusion-Limited Aggregation (or DLA), has a similar local growth law, but significantly different global behavior. For both LG and DLA, a proper description for the scaling properties of long-time solutions is not available yet. In this note, we outline a possible approach towards finding the correct theory yielding a regularized LG and its relation to DLA.

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