## 1 Large deviations and stochastic calculus (2004)

### BibTeX

@MISC{Guionnet041large,

author = {A. Guionnet},

title = {1 Large deviations and stochastic calculus},

year = {2004}

}

### OpenURL

### Abstract

Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, number theory, operator theory, quantum field theory, string theory etc... In the last ten years, they attracted lots of interests, in particular due to a serie of mathematical breakthroughs allowing for instance a better understanding of local properties of their spectrum, answering universality questions, connecting these issues with growth processes etc. In this survey, we shall discuss the problem of the large deviations of the empirical measure of Gaussian random matrices, and more generally of the trace of words of independent Gaussian random matrices. We shall describe how such issues are motivated either in physics/combinatorics by the study of the so-called matrix models or in free probability by the definition of a non-commutative entropy. We shall show how classical large deviations techniques can be used in this context. These lecture notes are supposed to be accessible to non probabilists and non freeprobabilists.

### Citations

658 |
Large Deviations Techniques and Applications
- Dembo, Zeitouni
- 1993
(Show Context)
Citation Context ...re devoted to the proof of large deviations principles, let us remind the reader what is a large deviation principle and the few main ideas which are commonly used to prove it. We refer the reader to =-=[41]-=- and [43] for further developments. In what follows, X will be a Polish space (that is a complete separable metric space). We then have Definition 2.1. • I : X → R + ∪ {+∞} is a rate function, iff it ... |

341 | On the distribution of the length of the longest increasing subsequence of random permutations
- Baik, Deift, et al.
- 1999
(Show Context)
Citation Context ... might provide an additional motivation to study determinantal laws. (1.0.1) Even more striking is the occurrence of large Gaussian matrices laws for the problem of the longest increasing subsequence =-=[8]-=-, directed polymers and the totally asymmetric simple exclusion process [75]. These relations are based on bijections with pairs of Young tableaux. In fact, the law of the hitting time of the totally ... |

193 | Spectral analysis of large dimensional random matrices, second edition - Bai, Silverstein - 2010 |

72 |
On the eigenvalues of random matrices
- Diaconis, Shahshahani
- 1994
(Show Context)
Citation Context ...uctuations to its natural non-commutative framework. They applied it with P. Sniady [97] to unitary matrices, generalizing to a non-commutative framework the results of P. Diaconis and M. Shahshahani =-=[37]-=- showing that traces of moments of unitary matrices converge towards Gaussian variables. In [60], I used the non-commutative framework to study fluctuations of the spectral measure of Gaussian band ma... |

60 |
Large deviations for Wigner’s law and Voiculescu’s noncommutative entropy, Prob
- Arous, Guionnet
- 1997
(Show Context)
Citation Context ...ical behavior in terms of large deviations in the cases listed above, with the restriction to Gaussian entries. They rely on a series of papers I have written on this subject with different coauthors =-=[10, 18, 29, 30, 42, 60, 62, 64, 65]-=- and try to give a complete accessible overview of this work to uninitiated readers. Some statements are improved or corrected and global introductions to free probability and hydrodynamics/large devi... |

53 | Free convolution of measures with unbounded support - Bercovici, Voiculescu - 1993 |

52 | Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free
- Collins
- 2003
(Show Context)
Citation Context ...thematical (or at least my) understanding. Haar distributed Unitary matrices also can be used to enumerate combinatorial objects due to their relation with representations of the symmetric group (c.f =-=[34]-=- for instance). Nice applications to the enumeration of magic squares can be found in [38]. In this domain, one tries to estimate integrals such as ZN(P), and in particular tries to obtain the full ex... |

41 | Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations
- Brenier
- 1999
(Show Context)
Citation Context ...nverse h = g−1 and Jacobian Jg. Note here that div(a g t (x)) = 0. Such an approach yields the Euler’s equation (5.2.4). The last way is to use convex analysis, following for instance Y. Brenier (see =-=[26]-=-, section 2). These two last strategies can only be applied when we know a priori that (µA,µB) are compactly supported (this indeed guarantees some a priori bounds on (ρ∗,u ∗ρ∗ ) for instance). When t... |

36 | Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix - Bai, Yin - 1993 |

35 | Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Probab. Theory Relat. Fields 112 - Biane, Speicher - 1998 |

33 |
Rectangular arrays with fixed margins, Discrete Probability and Algorithms
- Diaconis, Gangolli
- 1993
(Show Context)
Citation Context ...sed to enumerate combinatorial objects due to their relation with representations of the symmetric group (c.f [34] for instance). Nice applications to the enumeration of magic squares can be found in =-=[38]-=-. In this domain, one tries to estimate integrals such as ZN(P), and in particular tries to obtain the full expansion of log ZN(P) in terms of the dimension N. This could be done rigorously so far onl... |

32 |
Convergence rate of expected spectral distributions of large random matrices
- Bai
- 1993
(Show Context)
Citation Context ... is a common strategy to study the spectral measure of self-adjoint random matrices. This convergence can also be proved by considering the Stieljes transform of the spectral measure following Z. Bai =-=[4]-=-, which demands less hypothesis on the moments of the entries of X N . In the case of Gaussian entries, this result can be easily deduced from the large deviation principle of section 3. The convergen... |

32 | Eigenvalue distribution of large random matrices, from one matrix to several coupled matrices” Nucl. Phys - Eynard - 1997 |

32 |
Master loop equations, free energy and correlations for the chain of matrices
- Eynard
(Show Context)
Citation Context ...a|x| 2+ǫ + b with some a > 0 and ǫ > 0. In particular, we do not need any analyticity assumptions which can be required for instance to obtain the so-called Master loop equations (see Eynard and als. =-=[50, 49]-=-). Proof of Theorem 5.2 : It is enough to notice that, when diagonalizing the matrices Ai’s, the interaction is expressed in terms of spherical integrals by (3.1.1). Laplace’s (or saddle point) method... |

31 |
The Ising model on a random planar lattice: the structure of the phase transition and the exact critical exponents
- Boulatov, Kazakov
- 1987
(Show Context)
Citation Context ...though the free energy can still be defined in this domain, its description as a generating function becomes even more unclear. There is however some challenging works on this subject in physics (c.f =-=[21]-=- for instance). There are many other matrix models to be understood with nice combinatorial interpretations. A few were solved in physics literature by means of character expansions for instance in th... |

21 |
P.: Calcul stochastique non commutatif
- Biane
- 1995
(Show Context)
Citation Context ...rk to consider large random matrices. Finally, we will sketch the proofs of some results we needed in the previous chapter. 6.1 A few notions about von Neumann algebras Definition 6.1 (Definition 37, =-=[16]-=-). A C ∗ -algebra (A, ∗) is an algebra equipped with an involution ∗ and a norm ||.||A which furnishes it with a Banach space structure and such that for any X,Y ∈ A, ‖XY‖A ≤ ‖X‖A ‖Y‖A, ‖X ∗ ‖A = ‖X‖A... |

20 |
On the free convolution with a semi-circular distribution
- Biane
- 1997
(Show Context)
Citation Context ...∗ ( 3 t (x) ≤ 4π3t2 (1 − t) 2 )1 3 (x − x0) 1 3 if x0 is the nearest point of x in Ω c t. All these properties are due to the free convolution by the semi-circular law, and are direct consequences of =-=[15]-=-. Once Corollary 5.7 is given, the variational study of FIsing is fairly standard and gives Theorem 5.4. We do not detail this proof here. Hence, free probability arises naturally when we deal with th... |

20 |
Laplace approximations for sums of independent random vectors
- Bolthausen
- 1986
(Show Context)
Citation Context ... would be very tempting to try to generalize precise Laplace’s methods which are commonly used to understand the second order corrections of the free energy of mean field interacting particle systems =-=[20]-=-. However, such an approach until now failed even in the one matrix case due to the singularity of the logarithmic interacting potential (c.f. [33]). Another approach to this problem has recently been... |

20 |
Brownian motion model for the eigenvalues of a random matrix
- Dyson
- 1962
(Show Context)
Citation Context ...point to prove Theorem 4.5 is to observe that the evolution of ˆµ N is described, thanks to Itô’s calculus, by an autonomous differential equation. This is easily seen from the fact observed by Dyson =-=[45]-=- (see also [93], Thm 8.2.1) that the eigenvalues (λi t ,1 ≤ i ≤ N,0 ≤ t ≤ 1) of (Y N,β (t),0 ≤ t ≤ 1) are described as the strong solution of the interacting particle system dλ i t = √ 2 √ dB βN i 1 ∑... |

17 | Invariant subspaces of the quasinilpotent DToperator
- Dykema, Haagerup
(Show Context)
Citation Context ..., ˜ T are free copies of T. Hence, since δ(C) ≤ 2, we can hope that δ(T) < 2. However, based on an heavy computation of moments of these DT-operators due to P. Sniady [111], K. Dykema and U. Haagerup =-=[44]-=- could prove that T generates L(F 2 ). Hence invariance would be disproved if δ(T) < 2. But in fact, L. Aagaard [1] recently proved that δ ∗ (T) = 2 which shows at least that T is not a counterexample... |

12 |
On the convergence of the spectral empirical process of Wigner matrices
- Bai, Yao
(Show Context)
Citation Context ... of the spectral measure of Gaussian band matrices, following an idea of D. Shlyakhtenko [109]. On the other hand, A. Khorunzhy, B. Khoruzhenko and L. Pastur [89] and more recently Z. Bai and J.F Yao =-=[6]-=- developed Stieljes transforms technology to study the central limit theorems for entries with eventually only the four first moments bounded. Such techniques apply at best to prove central limit theo... |

12 |
1/N 2 corrections to free energy in Hermitian twomatrix model”, hep-th/0401166. 26 B.Eynard, “Loop equations for the semiclassical 2-matrix model with hard edges
- Eynard, Kokotov, et al.
(Show Context)
Citation Context ...er formula for the limiting spherical integral in the physics literature, but mostly saddle point studies of this a priori converging quantity. I do however mention recent works of B. Eynard and als. =-=[50]-=- and M. Bertola [14] who produced a formula of the free energy for the model of matrices coupled in chain by means of residues technology. However, this corresponds to the case where the matrices EN,D... |

11 | Large deviations from the circular law - Arous, Zeitouni - 1998 |

11 | Guionnet ; Large deviations upper bounds and non commutative entropies for some matrices ensembles
- Cabanal-Duvillard, A
- 1999
(Show Context)
Citation Context ...ical behavior in terms of large deviations in the cases listed above, with the restriction to Gaussian entries. They rely on a series of papers I have written on this subject with different coauthors =-=[10, 18, 29, 30, 42, 60, 62, 64, 65]-=- and try to give a complete accessible overview of this work to uninitiated readers. Some statements are improved or corrected and global introductions to free probability and hydrodynamics/large devi... |

10 |
McLaughlin: ”Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration
- Ercolani, T-R
(Show Context)
Citation Context ... full expansion of log ZN(P) in terms of the dimension N. This could be done rigorously so far only for one matrix models by use of Riemann-Hilbert problem techniques by J. Mc Laughlin et N. Ercolani =-=[46]-=-. First order asymptotics for a few several-matrix models could be obtained by orthogonal polynomial methods by M. L. Mehta [93, 90, 32] and by large deviations techniques in [61]. The physics literat... |

9 | Some remarks on Wigner distribution - Berezin - 1973 |

9 | Eigenvalue distribution of large random matrices with correlated entries - Khorunzhy - 1996 |

9 |
A method of integration over matrix variables
- Chadha, Mahoux, et al.
- 1981
(Show Context)
Citation Context ... of Riemann-Hilbert problem techniques by J. Mc Laughlin et N. Ercolani [46]. First order asymptotics for a few several-matrix models could be obtained by orthogonal polynomial methods by M. L. Mehta =-=[93, 90, 32]-=- and by large deviations techniques in [61]. The physics literature on the subject is much more consistent as can be seen on the arxiv (see work by V. Kazakov, I. Kostov, M. Staudacher, B. Eynard, P. ... |

7 |
Large deviations bounds for the law of the trajectories of the Hermitian Brownian motion Inventiones Mathematicae 152
- Biane, Capitaine, et al.
(Show Context)
Citation Context ...ical behavior in terms of large deviations in the cases listed above, with the restriction to Gaussian entries. They rely on a series of papers I have written on this subject with different coauthors =-=[10, 18, 29, 30, 42, 60, 62, 64, 65]-=- and try to give a complete accessible overview of this work to uninitiated readers. Some statements are improved or corrected and global introductions to free probability and hydrodynamics/large devi... |

7 |
Large deviations, Pure and
- Deuschel, Stroock
- 1989
(Show Context)
Citation Context ...d to the proof of large deviations principles, let us remind the reader what is a large deviation principle and the few main ideas which are commonly used to prove it. We refer the reader to [41] and =-=[43]-=- for further developments. In what follows, X will be a Polish space (that is a complete separable metric space). We then have Definition 2.1. • I : X → R + ∪ {+∞} is a rate function, iff it is lower ... |

6 | Finite free entropy and free group factors
- Brown
(Show Context)
Citation Context ... definition). It is currently under study whether δ is an invariant of the von Neumann algebra, that is if it satisfies (7.0.1). We note however that the converse implications is false since N. Brown =-=[27]-=- just produced an example showing that there exists a von-Neumann algebra which is not isomorphic to the free group factor but with same entropy dimension. Eventhough this problem has not yet been sol... |

6 | Diaconis: New tests of the correspondence between unitary eigenvalues and the zeros of Riemann’s zeta function - Coram, P |

5 |
Fluctuations de la loi spectrale des grandes matrices aléatoires
- Cabanal-Duvillard
- 1999
(Show Context)
Citation Context ...ments {N −1 Tr((X N ) p ),p ∈ N}. Such results where generalized to the non-commutative setting where one considers polynomial functions of several independent random matrices by T. Cabanal Duvillard =-=[28]-=- and myself [60]. Recently, J. Mingo and R. Speicher [96] gave a combinatorial interpretation of the limiting covariance via a notion of second order freeness which places the problem of fluctuations ... |

5 |
GUIONNET ; Discussions around non-commutative entropies
- CABANAL-DUVILLARD, A
- 2002
(Show Context)
Citation Context ...nothing to do with the existence and uniqueness of a strong solution of our free Fokker-Planck equation and we can indeed prove this result by means of R-transform theory for any ν ∈ {S 0,1 < ∞} (see =-=[30]-=-). 6.7 The infimum of SµD is achieved at a free Brownian bridge Let us state more precisely the theorem obtained in this section. A free Brownian bridge between µ0 and µ1 is the law of Xt = (1 − t)X0 ... |

4 |
Aging of spherical spin glasses, Prob
- Arous, Dembo, et al.
- 2001
(Show Context)
Citation Context ...artitions for instance in [40] for uniform distribution. Let us finally mention that large deviations can as well be obtained for the law of the largest eigenvalue of the Gaussian ensembles (c.f e.g. =-=[9]-=-, Theorem 6.2) ZN,β = (z N,β ij )1≤i,j≤N with (z N,β ij variables with covariance N −1c N,β i=1Chapter 4 Asymptotics of spherical integrals In this chapter, we shall consider the spherical integral I... |

4 | On the norm of random matrices - Monvel, Shcherbina |

4 | Girko ; Generalized Wigner law for band random matrices - Casati, V - 1993 |

3 |
An entropy approach to the time reversal of diffusion processes
- unknown authors
- 1985
(Show Context)
Citation Context ...sfies a large deviation principle with rate function given, for p ∈ C([0,1], P(R)), by S(p) = inf{I(µ|W) : (xt)#µ = pt ∀t ∈ [0,1]}. Here, (xt)#µ denotes the law of xt under µ. It was shown by Föllmer =-=[51]-=- that in fact S(p) is infinite unless there exists k ∈ L2 (pt(dx)dt) such that inf f∈C1,1 ∫ 1 ∫ (∂xf(x,t) − k(x,t)) (R×[0,1]) 2 pt(dx)dt = 0, (4.2.12) 034 and for all f ∈ C 2,1 (R × [0,1]), Moreover,... |

2 |
Thenon-microstates free entropy dimension of DT-operators. Preprint Syddansk Universitet
- AAGARD
- 2003
(Show Context)
Citation Context ...1 δ λi(A) ∈ P(R). For two Polish spaces X,Y we denote by C0 b (X,Y ) (or C(X,Y ) when no ambiguity is possible) the space of bounded continuous functions from X to Y . For instance, we shall denote C(=-=[0,1]-=-, P(R)) the set of continuous processes on [0,1] with values in the set P(R) of probability measures on R, endowed with its usual weak topology. For a measurable set Ω of R × [0,1], C p,q b (Ω) denote... |

2 | ZEITOUNI; A clt for a band matrix model, preprint - ANDERSON, O - 2004 |

2 |
KUIJLAARS ; Large n limit of Gaussian matrices with external source, part I
- BLEHER, A
(Show Context)
Citation Context ...tion and the limiting spectral distribution of A. A more detailed study based on Riemann-Hilbert techniques gives the limiting eigenvalue distribution correlations when ˆµ N A = αNδa +(1 −αN)δ−a (c.f =-=[19, 3]-=-). It is natural to wonder whether such a result could be derived from the Brownian paths description of the matrix. Our result allows (as a mild generalization of the next chapter) to describe the li... |

2 |
and third observables of the two-matrix model
- BERTOLA, Second
(Show Context)
Citation Context ...imiting spherical integral in the physics literature, but mostly saddle point studies of this a priori converging quantity. I do however mention recent works of B. Eynard and als. [50] and M. Bertola =-=[14]-=- who produced a formula of the free energy for the model of matrices coupled in chain by means of residues technology. However, this corresponds to the case where the matrices EN,DN of the spherical i... |

2 | The degree distribution in bipartite planar maps: applications to the Ising model http://front.math.ucdavis.edu/math.CO/0211070
- MELOU, SCHAEFFER
(Show Context)
Citation Context ... is the subject of the next section. Let us remark before embarking in this line of attack that a more direct combinatorial strategy can be developed. For instance, G. Scheaffer and M. Bousquet Melou =-=[22]-=- studied the Ising model on planar random graph as a generating function for the enumeration of colored planar maps, generalizing Tutte’s approach. The results are then more explicitly described by an... |

2 |
KHORUNZHI; On universality of the smoothed eigenvalue density of large random matrices
- MONVEL, A
- 1999
(Show Context)
Citation Context ...e law of the entries; this important field of investigation is often referred to as universality. An important effort of investigation was made in the last ten years in this direction for instance in =-=[23]-=-, [54], [76],[89], [110], [112], [118], [102] ... 3. Large random matrices and Riemann Zeta function : The Riemann Zeta function is given by ∞∑ ζ(s) = n −s n=1 with Re(s) > 1 and can be analytically c... |

2 |
A limit formula for a class of Gibbs measures with long range pair interactions
- CHIYONOBU
- 2000
(Show Context)
Citation Context ...ee energy of mean field interacting particle systems [20]. However, such an approach until now failed even in the one matrix case due to the singularity of the logarithmic interacting potential (c.f. =-=[33]-=-). Another approach to this problem has recently been proposed by B. Eynard et all[50, 49] and M. Bertola [14]. On a more analytic point of view, it would be interesting to understand better the prope... |

2 |
L 2 -Homology for von Neumann Algebras http://front.math.ucdavis.edu/math.OA/0309343
- CONNES, SHLYAKHTENKO
(Show Context)
Citation Context ... stands for the free convolution by m free semi-circular variables with parameter ǫ > 0. Define accordingly δ ∗ ,δ ∗∗ . Then δ ∗∗ (τ) ≤ δ(τ) ≤ δ ∗ (τ). In a recent work, A. Connes and D. Shlyaktenkho =-=[35]-=- defined another quantity ∆, candidate to be an invariant for von Neumann algebras, by generalizing the notion of L 2 -homology and L 2 -Betti numbers for a tracial von Neumann algebra. Such a definit... |

2 |
COMETS; Large deviations for random matrices and random graphs, preprint
- DEMBO
- 2004
(Show Context)
Citation Context ... a given spectral distribution. This problem seems to be very difficult in general. The deviations of a empirical moments of matrices with eventually non-centered entries of order N −1 are studied in =-=[39]-=- ; in this case, deviations are typically23 produced by the shift of all the entries and the scaling allows to see the random matrix as a continuous operator. This should not be the case for Wigner m... |

2 |
ZEITOUNI; Large deviations for integer paprtitions
- VERSHIK
(Show Context)
Citation Context ...ich are absolutely continuous with respect to Lebesgue measure and with density bounded by one. More general large deviations techniques have been developed to study random partitions for instance in =-=[40]-=- for uniform distribution. Let us finally mention that large deviations can as well be obtained for the law of the largest eigenvalue of the Gaussian ensembles (c.f e.g. [9], Theorem 6.2) ZN,β = (z N,... |

2 | ZEITOUNI; Moderate Deviations for the Spectral Measure of Random Matrices - DEMBO, GUIONNET, et al. |

2 | Random matrices, http://www-spht.cea.fr/cours-ext/fr/lectures notes.shtml - EYNARD |