Lecture notes and references

References may have various formats, depending on how they have been provided by the lecturers. Lecture notes have been numerized from hand-written notes.


Lecture notes (pdf, 44 Mo, by Martin Lenz)

References (archived directories)

  1. Extremes, Central Limit Theorem and records
  2. Rare events
  3. Power laws
  4. Random Energy Model, pinning and Burgers
  5. Directed polymers


Lecture notes (pdf, 12 Mo, by Patrik Ferrari)


References are written in the lecture notes.


Lecture notes (pdf, 53 Mo, by Martin Lenz)

Lecture notes (pdf, 2 Mo, by Vivien Lecomte [less neat])


Functional Renormalization for Disordered Systems, Basic Recipes and Gourmet Dishes
K.J. Wiese and P. Le Doussal, Markov Processes Relatd Fields, Vol 13 (2007) 777 (arXiv:condmat/0611346).

Exact Results and open questions in first principle functional RG
P. Le Doussal, arXiv:condmat/0809.1192 and refs therein.


Lecture notes (pdf, 36 Mo, by Martin Lenz)


1. EVS for UNCORRELATED random variables: the three limiting distributions Gumbel, Frechet and Weibull

(i) "Statistics of Extremes", E.J. Gumbel (Columbia Univ., New York, 1958).
(ii) "The Asymptotic Theory of Extreme Order Statistics", J. Galambos (Newyork, 1978).
(iii) for a recent book with nice historical surveys, see "EXTREME VALUE DISTRIBUTIONS Theory and Applications" by Samuel Kotz (The George Washington University, USA) & Saralees Nadarajah (The University of Nottingham, UK), see http://www.worldscibooks.com/mathematics/p191.html (see especially chapter 1).
(iv) see also, J.-P. Bouchaud and M. Mezard, J. Phys. A 30, 7997 (1997).
(v) For a recent review on extreme statistics and its connection to travelling fronts, see S.N. Majumdar and P.L. Krapivsky, Physica 318A, 161 (2003). For applications of extreme value problems in computer science see "Understanding Search Trees via Statistical Physics", S.N. Majumdar, D.S. Dean and P.L. Krapivsky, proceedings of the STATPHYS-2004, arXiv:cond-mat/0410498.
(vi) The same three limiting extreme distributions also appear in other problems, see for example two recent articles:
(a) "Density of Near Extreme Events", S. Sabhapandit and S.N. Majumdar, PRL, 98, 140201 (2007).
(b) "Level Density of Bose Gas and extreme Value Statistics", A. Comtet, P. Leboeuf, and S.N. Majumdar, PRL, 98, 070404 (2007).

2. EVS for CORRELATED random variables: simple examples

In the lectures I discussed different methods to compute the first-passage statistics for Brownian motion including the path-integral methods. For the standard Fokker-Planck methods (forward and backward) see the books:
(i) H. Risken, "The Fokker-Planck Equation: Methods of Solutions and Applications", 2nd edition, Springer Series in Synergetics, Springer.
(ii) S. Redner, "A Guide to first-passage processes"
But they do include neither the path-integral methods nor the Feynman-Kac formula. For quick learning of these methods (with the help of several examples), I refer to my recent brief review article:
"Brownian Functionals in Physics and Computer Science", S.N. Majumdar, current science, vol-89, 2076 (2005). This is also available on the web: arXiv:cond-mat/0510064.
In this article you will also find references to many of the original articles and other articles in this field.

3. Variety of CONSTRAINED Brownian motions

For the maximum of various constrained Brownian motions, I suggest the excellent short review (essay) by S. Finch, "Variants of Brownian Motion" availbale at: http://algo.inria.fr/csolve/br.pdf (actually his home page contains many beatiful essays, strongly recommended). This will also contain references to the original articles.
For the time at which the maximum occurs in a variety of constrained Brownian motions I suggest the two following recent articles and references therein :
(i) "Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time", J. Randon-Furling and S.N. Majumdar, JSTAT, P10008 (2007).
(ii) "On the time to reach maximum for a variety of constrained Brownian motions", S.N. Majumdar, J. Randon-Furling, M.J. Kearney and M. Yor, arXiv:0708.2101.
For the Airy-distribution function and its various applications, here are some references:
(i) P. Flajolet, P. Poblete, and A. Viola, "On the analysis of linear probing with hashing", Algorithmica 22, 490 (1998) (this article reviews many problems in computer science and graph theory where the Airy-distribution appears).
(ii) L. Takacs, "A Bernoulli excursion and its various applications", Adv. in Appl. probab. 23, 557 (1991).
(iii) See also a very nice recent reviews by S. Janson, "Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas", Probability Surveys 3 (2007), 80-145. and also, G. Louchard and S. Janson, "Tail estimates for the Brownian excursion area and other Brownian areas", Electronic J. Probab. 12 (2007), no. 58, 1600-1632.
The path integral derivation of the area under the Brownain excursion that I discussed in the lectures can be found in:
(a) S.N. Majumdar and A. Comtet, "Airy Distribution Function: From the Area under a Brownain Excursion to the Maximal Height of Fluctuating Interfaces", J. Stat. Phys, 119, 777 (2005).
(b) The result that the Airy-distribution function also describes the maximum of the relative height for fluctuating interfaces in stationary states first appeared in, S.N. Majumdar and A. Comtet, "Exact maximal height distribution of fluctuating interfaces" PRL, 92, 225501 (2004).

4. Random Matrices: general introduction to random matrices

The best reference here is of course the famous book:
"Random Matrices", M.L. Mehta (second edition).
For the distribution of the maximum eigenvalue, see:
C.A. Tracy and H. Widom, "Level-Spacing distributions and the Airy Kernel", Commun. Math. Phys. 159, 151 (1994). After that there have been numerous works on this subject by Tracy-Widom as well as Baik, Rains, Johansson, Prahoffer, Spohn, Ferrari, Sassomoto and many others. For a quick summary of the three types of distributions I recommend:
(a) C.A. Tracy and H. Widom, "The Distribution of the Largest Eigenvalue in the Gaussian Ensembles: \beta=1,2,4" arXiv:solv-int/9707001.
(b) J. Baik and E.M. Rains, "Limiting distributions for a polynuclear growth model with external sources", arXiv:math.PR/0003130.
See also "Extreme value problems in Random Matrix Theory and other disordered systems" by G. Biroli, J.-P. Bouchaud and M. Potters, in JSTAT, proceedings of `Principles of Dynamics of Nonequilibrium Systems', ISI Cambridge 2006 (arXiv:cond-mat/0702244).

5. LARGE DEVIATIONS of the maximum eigenvalue: probability of RARE fluctuations

For large deviations of the density of states:
(i) G. Ben Arous and A. Guionnet, "Large deviations for Wigner's law and voiculescu's Non-commutative Entropy", Prob. The. Rel. Fields, 108, 517 (1997).
The general scaling of the large deviation of the maximum eigenvalue follows from the above paper (see also, K. Johansson, "Shape fluctuations and Random Matrices" arXiv:math.CO/9903134).
The explicit form of the left large deviation function of the maximal eigenvalue for Gaussian random matrices was first derived in:
D.S. Dean and S.N. Majumdar, "Large Deviations of Extreme Eigenvalues of Random Matrices", PRL, 97, 160201 (2006).
For detailed derivation (including the derivation of the Dyson formalism and the use of Tricomi's theorem in presence of a wall) see:
D.S. Dean and S.N. Majumdar, "Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices", Phys. Rev. E, 77, 041108 (2008).
To learn about Tricomi's theorem I recommend:
F.G. Tricomi, "Integral Equations" (London, 1957). See also the book "Integral Equations" by S.G. Mikhlin (Pergamon, London, 1957) (page 124-133).

6. WISHART Random matrices

On Wishart ensembles, there are numerous references. I suggest to look at the introduction of our recent paper for all references:
P. Vivo, S.N. Majumdar, and O. Bohigas, "Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices", J. Phys. A. Math. Theor. 40(16) (2007) 4317-4337.
For minimum eigenvalue, I suggest e.g.: A. Edelman, "Eigenvalues and condition numbers of random matrices", SIAM J. Matrix Anal. appl. 9, 543 (1988).
For applications of Wishart random matrices in QUANTUM ENTANGLEMENT problem, see(and refs therein):
S.N. Majumdar, O. Bohigas, and A. Lakshminarayan, "Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State", J. Stat. Phys. 131, 33 (2008).

7. UBIQUITY of the Tracy-Widom law

This is the part I did not have time to cover in the school. Tracy-Widom law apppears in many different problems. There are many reviews on this. I gave a set of lectures at Les Houches in 2006 where many references to the original articles can be found:
S.N. Majumdar, "Random Matrices, the Ulam Problem, Directed Polymers & Growth Models, and Sequence Matching", Les Houches lecture notes for the summer school on "Complex Systems" (Les Houches, July, 2006 organized by J.-P. Bouchaud and M. Mezard): arXiv:cond-mat/0701193.