Universal Randomness

During last two decades it has been discovered that the statistical properties of a number of microscopically rather different random systems at the macroscopic level are described by {\it the same} universal probability distribution function which is called the Tracy-Widom (TW) distribution. Among these systems we find both purely methematical problems, such as the longest increasing subsequences in random permutations, and quite physical ones, such as directed polymers in random media or polynuclear crystal growth. In the extensive Introduction we discuss in simple terms these various random systems and explain what the universal TW function is. Next, concentrating on the example of one-dimensional directed polymers in random potential we give the main lines of the formal proof that fluctuations of their free energy are described the universal TW distribution. The second part of the review consist of detailed appendices which provide necessary self-contained mathematical background for the first part.Comment: 34 pages, 6 figure

Replica solution of the Random Energy Model

The alternative replica technique which involve summation over all integer momenta of the partition function and which does not require analytic continuation to non-integer values of the replica parameter $n$ is discussed. In terms of this technique (which does not involve any replica symmetry breaking "magic operations") rigorous solution for the average free energy of the Random Energy Model is recovered in a very simple way.Comment: 8 page

Distribution function of the endpoint fluctuations of one-dimensional directed polymers in a random potential

The explicit expression for the the probability distribution function of the endpoint fluctuations of one-dimensional directed polymers in random potential is derived in terms of the Bethe ansatz replica technique by mapping the replicated problem to the N-particle quantum boson system with attractive interactions.Comment: 15 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1209.360