88 research outputs found
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 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
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
Two-temperature statistics of free energies in (1+1) directed polymers
The joint statistical properties of two free energies computed at two
different temperatures in {\it the same sample} of directed polymers is
studied in terms of the replica technique. The scaling dependence of the
reduced free energies difference
on the two temperatures and is derived. In particular, it is
shown that if the two temperatures are close to each
other the typical value of the fluctuating part of the reduced free energies
difference is proportional to . It is also
shown that the left tail asymptotics of this free energy difference probability
distribution function coincides with the corresponding tail of the TW
distribution.Comment: 7 page
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