Mathematics – Probability
Scientific paper
2007-11-08
Mathematics
Probability
Thesis, submitted to Indian Statistical Institute, Kolkata, India
Scientific paper
In this thesis, we consider several Random Energy Models. This includes Derrida's Random Energy Model (REM) and Generalized Random Energy Model (GREM) and a nonhierarchical version (BK-GREM) by Bolthausen and Kistler. The limiting free energy in all these models along with Word GREM, a model proposed by us, turn out to be a cute consequence of large deviation principle (LDP). This LDP argument allows us to consider non-Gaussian driving distributions as well as external field. We could also consider random trees as the underlying tree structure in GREM. In all these models, as expected, limiting free energy is not 'universal' unlike the SK model. However it is 'rate specific'. Consideration of non-Gaussian driving distribution as well as different driving distributions for the different levels of the underlying trees in GREM leads to interesting phenomena. For example in REM, if the Hamiltonian is Binomial with parameter $N$ and $p$ then the existence of phase transition depends on the parameter $p$. More precisely, phase transition takes place only when $p>{1/2}$. For another example, consider a 2 level GREM with exponential driving distribution at the first level and Gaussian in the second with equal weights at both the levels. Then even if the limiting ratio for the second level particles, $p_2$ is 0.00001 (very small), the system reduces to a Gaussian REM. On the other hand, if we consider a 2 level GREM with Gaussian driving distribution at the first level and exponential in the second, the system will never reduce to a Gaussian REM. In either case, the system will never reduce to that of an exponential REM. etc.
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