## ZOOMING IN ON LOCAL LEVEL STATISTICS BY SUPERSYMMETRIC EXTENSION OF FREE PROBABILITY (908)

### BibTeX

@MISC{Zirnbauer908zoomingin,

author = {M. R. Zirnbauer},

title = {ZOOMING IN ON LOCAL LEVEL STATISTICS BY SUPERSYMMETRIC EXTENSION OF FREE PROBABILITY},

year = {908}

}

### OpenURL

### Abstract

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### Citations

607 |
Orthogonal polynomials
- Szegö
- 1959
(Show Context)
Citation Context ...(x)dx , (3.21) πN−1,N(x) = Z −1 N ∫ R ∏ RN−1 2≤i< j (xi − x j) 2 ∏ 2≤l≤N (x − xl)e −NV(xl) dxl . (3.22) The function πN−1,N(x) is a polynomial of degree N − 1 in the variable x. By a classical result =-=[1, 19]-=- it is actually the orthogonal polynomial of degree N − 1 associated with the weight function e −NV(x) . We find it convenient to choose the normalization constants C3,N and ZN in such a way that πN−1... |

324 |
Heat Kernels and Dirac Operators
- Berline, Getzler, et al.
- 1992
(Show Context)
Citation Context ...u is a UN-invariant measure for CP N−1 . Now CP N−1 is a Kähler manifold with UN-invariant Riemannian geometry, and the function µ : CP N−1 → LieUN , g ·(U1 × UN−1) ↦→ igΠg −1 , is a momentum mapping =-=[3]-=-. We observe that the expression k Tr(XgΠg −1 ) in the exponent of our integrand is obtained by contracting µ with the Lie algebra element −ikX ∈ LieUN . It follows that the integral is governed by th... |

316 |
Free random variables
- Voiculesku, Dykema, et al.
- 1992
(Show Context)
Citation Context ...ic scattering. 1. Introduction The notions of ‘free probability’ and ‘freeness’ of non-commutative random variables were introduced by Voiculescu in the study of certain algebras of bounded operators =-=[24]-=-. The word freeness in this context means a kind of statistical independence of operators. The algebraic concept of freeness of random variables has a natural realization by random matrices in the lim... |

308 |
Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert Approach
- Deift
- 1999
(Show Context)
Citation Context ...du = C1,N k ∑ i=1 ekxi , (3.9) ∏ j(̸=i)(xi − x j) 3.3. Coulomb gas. — Based on the exact expressions (3.7)–(3.9), our goal is to compute the large-N asymptotics of N−1 lnΩ(NkΠ). We begin by recalling =-=[7]-=- that, governed by ∏i< j(xi −x j) 2 ∏l e−NV(xl) dxl , the eigenvalues x1,...,xN distribute for N → ∞ according to the equilibrium probability measure, ν, which is determined by minimizing Dyson’s Coul... |

305 |
The classical groups
- Weyl
- 1946
(Show Context)
Citation Context ...j1... jn = ∑ i ′ 1 ,...,i′ n , j′ 1 ,..., j′ n g i1 i ′ ···g 1 in i ′ n C i′ 1 ...i′ n j ′ 1 ... j′ n (g−1) j′ 1 j1 ···(g−1 ) j′ n jn (g ∈ GLN) . By a classical result of invariant theory due to Weyl =-=[26]-=-, one then knows that C i1...in j1... jn is nonzero only if the numbers { j1,..., jn} agree as a set with the set {i1,...,in}, i.e., if there exists an element π of the symmetric group Sn such that jl... |

214 |
Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory
- Deift, Kriecherbauer, et al.
- 1999
(Show Context)
Citation Context ...] from the upper or lower half of the complex plane, one rewrites the relation (3.11) as V ′ (x) = g+(x)+g−(x) . As it stands, this equation holds only for x ∈ [a,b] ⊂ R. However, from general theory =-=[8]-=- one knows that g±(x) are the two branches of a double-valued complexanalytic function z ↦→ (g(z),h(z)) evaluated at z = x. Thus by the principle of analyticFREE PROBABILITY MEETS SUPERSYMMETRY 13 co... |

175 | Transcending Classical Invariant Theory - Howe - 1989 |

139 |
Introduction to superanalysis
- Berezin
- 1987
(Show Context)
Citation Context ...gonal saddle Q0 = diag(g+(z),g−(z),g+(z),g−(z)) = Reg+(z)Id 2|2 + iImg+(z)s . The dominant part of the integrand is invariant under conjugation Q → T QT −1 by elements T of the Lie supergroup U 1,1|2 =-=[2]-=-. The orbit generated by the action on Q0 of this symmetry group is a supermanifold of saddle points Q = T Q0(z)T −1 = Reg+(z)Id 2|2 + iImg+(z)TsT −1 .34 S. MANDT AND M.R. ZIRNBAUER Evaluating the do... |

138 | Limit laws for random matrices and free products - Voiculescu - 1991 |

120 |
Addition of certain non-commutative random variables
- Voiculescu
- 1986
(Show Context)
Citation Context ...present paper. A central tool of the free probability formalism is the so-called R-transform, which resembles the logarithm of the characteristic function for commutative random variables. Voiculescu =-=[22]-=- defines it by the functional inverse of the average trace of the resolvent operator. A second approach to the subject is due to Speicher [18], who expresses the moments of the random matrix directly ... |

105 |
Multiplicative functions on the lattice of non-crossing partitions and free
- Speicher
- 1994
(Show Context)
Citation Context ...tic function for commutative random variables. Voiculescu [22] defines it by the functional inverse of the average trace of the resolvent operator. A second approach to the subject is due to Speicher =-=[18]-=-, who expresses the moments of the random matrix directly in terms of the Taylor coefficients of the R-transform. Speicher’s concept of non-crossing partition is a mathematical expression of the domin... |

53 | Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free
- Collins
(Show Context)
Citation Context ...fined to a finite interval [a,b] and assume that the empirical measure N−1 ∑ j δ(x − x j,N) converges weakly to a measure with support in [a,b]. Under these conditions the following is known. Collins =-=[6]-=- differentiates the scaled logarithm of the spherical integral n times at zero to show that lim N→∞ N−1 dn dk ln IXN (Nk) ∣ n ∣ k=0 = (n − 1)!cn , (1.6) i.e., he establishes convergence to the nth fre... |

42 | Random matrices: universality of local eigenvalue statistics
- Tao, Vu
(Show Context)
Citation Context ...and easily extend to ensembles of different symmetry type. Let us finish with a quick glance at a new and exciting development. In a long series of papers by Erdös, Ramirez, Schlein, Yau, and Tao, Vu =-=[10, 11, 20, 12, 13]-=-, sine-kernel (or GUE) universality of spectral correlations has recently been established for the case of Hermitian Wigner matrices, i.e., random matrices with statistically independent entries. Usin... |

30 | Universality of correlation functions of Hermitian random matrices in an external field
- Zinn-Justin
- 1998
(Show Context)
Citation Context ... are now going to study Ω(K) for the rank-one case K = kΠ with k ∈ R and Π the projector on a one-dimensional subspace of C N . A related situation has been investigated in the work of P. Zinn-Justin =-=[28, 29]-=-, Collins [6], and Guionnet & Maida [14]; we will comment on the literature as we go along. We begin by reviewing some basic material. 3.1. Voiculescu R-transform. — As before, let µN be a probability... |

25 | E.: Products and ratios of characteristic polynomials of random Hermitian matrices
- Baik, Deift
- 2003
(Show Context)
Citation Context ...(x)dx , (3.21) πN−1,N(x) = Z −1 N ∫ R ∏ RN−1 2≤i< j (xi − x j) 2 ∏ 2≤l≤N (x − xl)e −NV(xl) dxl . (3.22) The function πN−1,N(x) is a polynomial of degree N − 1 in the variable x. By a classical result =-=[1, 19]-=- it is actually the orthogonal polynomial of degree N − 1 associated with the weight function e −NV(x) . We find it convenient to choose the normalization constants C3,N and ZN in such a way that πN−1... |

22 | A Fourier view on the R-transform and related asymptotics of spherical integrals
- Guionnet, Maïda
- 2005
(Show Context)
Citation Context ...hes convergence to the nth free cumulant (times a factorial). A stronger version of this result, lim N→∞ N−1 ln IXN (Nk) = ∫ k 0 R(k ′ )dk ′ = ∞ cn ∑ n=1 n kn , (1.7) was proved by Guionnet and Maida =-=[14]-=- under the condition that k ∈ C is small enough. (Notice that (1.7) implies (1.6).) For k real and large, however, the authors of [14] obtain a different behavior, separated from the small-k regime by... |

20 |
Supersymmetry and theory of disordered metals
- Efetov
(Show Context)
Citation Context ...nd, we now change subject and turn to the so-called supersymmetry technique, by which we mean the method of integration over commuting and anti-commuting variables pioneered by Wegner [25] and Efetov =-=[9]-=-. In its original formulation (using the Hubbard-Stratonovich transformation) this method was limited to Gaussian disorder distributions. Nonetheless, with this limitation it has enjoyed great success... |

17 | Bulk universality for Wigner matrices
- Erdős, Péché, et al.
- 2010
(Show Context)
Citation Context ...and easily extend to ensembles of different symmetry type. Let us finish with a quick glance at a new and exciting development. In a long series of papers by Erdös, Ramirez, Schlein, Yau, and Tao, Vu =-=[10, 11, 20, 12, 13]-=-, sine-kernel (or GUE) universality of spectral correlations has recently been established for the case of Hermitian Wigner matrices, i.e., random matrices with statistically independent entries. Usin... |

15 | H.-T.: Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation
- Erdős, Ramirez, et al.
- 2010
(Show Context)
Citation Context ...and easily extend to ensembles of different symmetry type. Let us finish with a quick glance at a new and exciting development. In a long series of papers by Erdös, Ramirez, Schlein, Yau, and Tao, Vu =-=[10, 11, 20, 12, 13]-=-, sine-kernel (or GUE) universality of spectral correlations has recently been established for the case of Hermitian Wigner matrices, i.e., random matrices with statistically independent entries. Usin... |

15 |
The mobility edge problem: Continuous symmetry and a conjecture, Zeitschrift für
- Wegner
- 1979
(Show Context)
Citation Context ...aring this in mind, we now change subject and turn to the so-called supersymmetry technique, by which we mean the method of integration over commuting and anti-commuting variables pioneered by Wegner =-=[25]-=- and Efetov [9]. In its original formulation (using the Hubbard-Stratonovich transformation) this method was limited to Gaussian disorder distributions. Nonetheless, with this limitation it has enjoye... |

13 | Yau Bulk universality for Wigner Hermitian matrices with subexponential decay
- Erdős, Ramirez, et al.
(Show Context)
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12 | H.-T.: Universality of random matrices and local relaxation flow
- Erdős, Schlein, et al.
(Show Context)
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11 |
Lie groups, Graduate Texts
- Bump
(Show Context)
Citation Context ...4.3). For this we start from the relation = s λ(K)s λ(H) s λ(IdN) , (4.4) Det(IdN − K) = ∑ N m=0 (−1)m s [1 m ](K) , (4.5) which is a special case of what is sometimes called the dual Cauchy identity =-=[5]-=-. Then, substituting K = ψ ⊗ ˜ψ we manipulate the left-hand side as follows: Det(IdN − ψ ⊗ ˜ψ) = e Tr ln(IdN−ψ⊗˜ψ) −∑ = e N m=1 m−1Tr(ψ⊗˜ψ) m = e +∑Nm=1 m−1 〈˜ψ,ψ〉 m = e −ln(1−〈˜ψ,ψ〉) = (1 − 〈˜ψ,ψ〉) −... |

4 |
Adding and multiplying random matrices: a generalization of Voiculescu’s formulas
- Zinn-Justin
(Show Context)
Citation Context ...an one. Since their inception in the 1980’s, free probability theory and the supersymmetry method have coexisted with little or no mutual interaction. Forecast in a prescient remark by P. Zinn-Justin =-=[29]-=-, our message is that a new quality emerges when the two formalisms are combined. More specifically, we will show that the characteristic function of the random matrix ensemble – an object of central ... |

3 | Superbosonization of invariant random matrix ensembles
- Littelmann, Sommers, et al.
- 2008
(Show Context)
Citation Context ... G-actions on V resp. ∧(V ∗ ). Later we will write this equivariance property as Ω(K) = Ω(gKg−1 ) for short. In such a setting, the superbosonization method offers a reduction step which is available =-=[16]-=- for the classical Lie groups G = UN , ON , and USpN . For each of these groups, the algebra of G-equivariant differential forms on V is generated by (the dual of) the Z2-graded vector space W = W0 ⊕ ... |

3 |
of addition in random matrix theory
- Zee
- 1996
(Show Context)
Citation Context ...the R-transform is closely related to the logarithm of the Fourier transform of µN , N → ∞. Let us mention in passing that, since g(z) is also known as the Green’s function, physicists by Zee’s fancy =-=[27]-=- sometimes call b(k) := k−1 + R(k) the Blue’s function. 3.1.1. Examples. — The Gaussian measure µN with density dµN(H) ∝ e− N 2 TrH2 dH is called the Gaussian Unitary Ensemble (GUE) with c2 = 1. For t... |

2 |
Grassmann variables in stochastic quantum physics: the case of compound-nucleus scattering, Phys. Rep
- Verbaarschot, Weidenmüller, et al.
- 1985
(Show Context)
Citation Context ...we know that the equation q −1 + R(q) = z for a scalar variable q ∈ C has two solutions, q = g+(z) and q = g−(z), with Reg+(z) = Reg−(z) , Img+(z) = −Im g−(z) < 0 . The following analysis is standard =-=[21]-=- and we therefore give only a sketch. We first look for diagonal matrices Q that solve the saddle-point equation (6.9). The condition isQ ˜ϕϕ > 0 selects Q ˜ϕϕ = diag(g+(z),g−(z)) . (This is literally... |

1 | Energy correlations for a random matrix model of disordered bosons - Lueck, Sommers, et al. - 2006 |