large nc confinement, universal shocks, and random matrices
TRANSCRIPT
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Large Nc confinement, universal shocks, and random
matrices
Jean-Paul Blaizot, IPhT- Saclay
EMMI - Workshop
February 19, 2009
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Outline
- Phase transition in large Nc Yang-Mills theories, in d=2 (Durhuus-Olesen), possible universality and simulations in d=3,4 (Narayanan-Neuberger)
-(Fluid)-dynamics of eigenvalues of Wilson loops, (complex) Burgers equation - Dyson’s fluid
- Further developments : finite Nc effects as viscosity corrections - relation with random matrix theory.
Work done in collaboration with Maciej Nowak(arXiv: 0801.1859 - PRL 101:102001 (2008)and arXiv: 0902.2223)
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Small loops - Short distance physics - weak interactions
large loops
long distance physics
strong interactions Cross-over
Becomes a sharp transition as
“Confinement-deconfinement” transition (?)
Universal properties in d=2,3,4 (?)
The Wilson loop in SU(N) gauge theories
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Durhuus-Olesen phase transition(P. Durhuus and P. Olesen, NPB 1981)
Spectral density as a function of the area A (d=2)
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The moments of the spectral density
The moments are related to Laguerre polynomials
Remarkable properties at large n
Much studied
Olesen, Durhuus, Rossi, Kazakov, Douglas, Gross, Gopakumar, Matytsin, Migdal, etc.
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(R. Janick and W. Wieczorek, math-ph/0312043)
A random matrix perspective
A model for the Wilson loop
Resolvent and spectral density
Evolution equation
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Evolution from small areas to large areas (1)
(from R. Janick and W. Wieczorek, math-ph/0312043)
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Evolution from small areas to large areas (2)
(from R. Janick and W. Wieczorek, math-ph/0312043)
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Closing of the gap is universal
(R.Narayanan and H.Neuberger, arXiv:0711.4551)
Numerical studies on the lattice
Double scaling limit
Average characteristic polynomial
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(Fluid) dynamics of eigenvalues and the Burgers equation
Define
(Hilbert transform)
The function obeys the Burgers equation
(Olesesn, Gross, Gopakumar, etc)
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Navier-Stokes equation
Euler equation
Burgers equation and equations of fluid dynamics
Important: F is here complex
Burger=s equation is often used as a model for turbulence (shocks)
(Hopf-Burgers eqn, free random variable calculus, Voiculescu …)
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Solution of the complex Burgers equation
Characteristics (= Lines of constant ‘velocity’)
Singularity when
Can be solved with Characteristics
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The qualitative behavior of the solution is determined by the location of the singularities in the complex plane
Asymptotic behavior of moments
Expanding near a singularity
One can invert the characteristic equation and get
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Gapped phase
Characterisics
singularity
Vicinity of the singularity
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Closing the gap
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Gapless phase
In vicinity of the singularity
Start with initial conditions
singularity
In the second case, the singularity hits the real axis in a finite time A1
Two cases
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〈δxi〉 = E(xi)∆t 〈(δxi)2〉 = ∆t
E(xj) =∑
i!=j
(1
xj − xi
)
∂P
∂t=
12
∑
i
∂2P
∂x2i
−∑
i
∂
∂xi(E(xi)P )
P (x1, · · · , xN , t) = C∏
i<j
(xi − xj)2 e−P
i
x2i
2t
Approaching
Dyson’s Brownian motion (hermitian matrices)
Fokker-Planck equation for the joint probability
Whose solution reads
(arXiv 0902.2223 and work in progress)
P (x1, · · · , xN , t)
(1)
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ρ̃(x, t) =∫ N∏
k=1
dxk P (x1, · · · , xN , t)N∑
l=1
δ(x− xl)
∂ρ̃(x, t)∂t
=12
∂2ρ̃(x, t)∂λ2
− ∂
∂λPV
∫dy
ρ̃(x, y, t)x− y
ρ̃(x, y) = ρ̃(x)ρ̃(y) + ρ̃con(x, y)
ρ̃(x) = Nρ(x)
∂ρ(x)∂τ
+∂
∂xρ(x)PV
∫dy
ρ(y)x− y
=1
2N
∂2ρ(x)∂x2
+PV∫
dyρcon(x, y)
x− y
τ = Nt
Approaching
Average density of eigenvalues (“one-particle density”)
Infinite hierarchy of equations
To study the large N limit, rescale
and get
(2)
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G(z, τ) =⟨
1N
Tr1
z −H(τ)
⟩=
∫dy
ρ(y, τ)z − y
∂τG(z, τ) + G(z, τ) ∂zG(z, τ) = 0
ρ(x, τ)
G(z, τ) =⟨
1N
Tr1
z −H(τ)
⟩=
∂
∂z
⟨1N
Tr ln (z −H(τ))⟩
=∂
∂z
⟨1N
ln det (z −H(τ))⟩
F (z, τ) =∂
∂z
1N
ln 〈det (z − H(τ))〉
〈det (z − H(τ))〉 =N∏
i=1
(z − x̄i)
F (z, τ) ≈ G(z, τ) as N →∞
∂τF (z, τ) + F (z, τ) ∂zF (z, τ) = − 12N
∂2zF (z, τ)
Approaching (3)
Resolvent
Equation for reduces to (inviscid) Burgers eqn. for G (in large N limit)
F fulfills the Viscid Burgers equation(EXACTLY !)
Average of the characteristic polynomial
(see also Neuberger, arXivO806.0149 , 0809.1238)
Note that
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Conclusions
- Many features of the large Nc transition are coded in the solution of a Simple
Burgers equation (universal shocks, etc). - Provides a simple understanding for the remarkable universality that is emerging from lattice calculations
- Finite Nc corrections appears as « viscous » effects in the fluid of eigenvalues
- A general picture emerges in the framework of random matrix theory