312,999 research outputs found
Probability distributions for directed polymers in random media with correlated noise
The probability distribution for the free energy of directed polymers in
random media (DPRM) with uncorrelated noise in dimensions satisfies the
Tracy-Widom distribution. We inquire if and how this universal distribution is
modified in the presence of spatially correlated noise. The width of the
distribution scales as the DPRM length to an exponent , in good (but not
full) agreement with previous renormalization group and numerical results. The
scaled probability is well described by the Tracy-Widom form for uncorrelated
noise, but becomes symmetric with increasing correlation exponent. We thus find
a class of distributions that continuously interpolates between Tracy-Widom and
Gaussian forms
The right tail exponent of the Tracy-Widom-beta distribution
The Tracy-Widom beta distribution is the large dimensional limit of the top
eigenvalue of beta random matrix ensembles. We use the stochastic Airy operator
representation to show that as a tends to infinity the tail of the Tracy Widom
distribution satisfies P(TW_beta > a) = a^(-3/4 beta+o(1)) exp(-2/3 beta
a^(3/2))
On the average of the Airy process and its time reversal
We show that the supremum of the average of the Airy process and its time
reversal minus a parabola is distributed as the maximum of two independent GUE
Tracy-Widom random variables. The proof is obtained by considering a directed
last passage percolation model with a rotational symmetry in two different
ways. We also review other known identities between the Airy process and the
Tracy-Widom distributions.Comment: 12 page
Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
We study a system of N non-intersecting Brownian motions on a line segment
[0,L] with periodic, absorbing and reflecting boundary conditions. We show that
the normalized reunion probabilities of these Brownian motions in the three
models can be mapped to the partition function of two-dimensional continuum
Yang-Mills theory on a sphere respectively with gauge groups U(N), Sp(2N) and
SO(2N). Consequently, we show that in each of these Brownian motion models, as
one varies the system size L, a third order phase transition occurs at a
critical value L=L_c(N)\sim \sqrt{N} in the large N limit. Close to the
critical point, the reunion probability, properly centered and scaled, is
identical to the Tracy-Widom distribution describing the probability
distribution of the largest eigenvalue of a random matrix. For the periodic
case we obtain the Tracy-Widom distribution corresponding to the GUE random
matrices, while for the absorbing and reflecting cases we get the Tracy-Widom
distribution corresponding to GOE random matrices. In the absorbing case, the
reunion probability is also identified as the maximal height of N
non-intersecting Brownian excursions ("watermelons" with a wall) whose
distribution in the asymptotic scaling limit is then described by GOE
Tracy-Widom law. In addition, large deviation formulas for the maximum height
are also computed.Comment: 37 pages, 4 figures, revised and published version. A typo has been
corrected in Eq. (10
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