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

- Extremes, Central Limit Theorem and records
- Rare events
- Power laws
- Random Energy Model, pinning and Burgers
- Directed polymers

References are written in the lecture notes.

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.

*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.