A realization of the Lie algebra associated to a Kantor triple system

We present a nonlinear realization of the 5-graded Lie algebra associated to a Kantor triple system. Any simple Lie algebra can be realized in this way, starting from an arbitrary 5-grading. In particular, we get a unified realization of the exceptional Lie algebras f_4, e_6, e_7, e_8, in which they are respectively related to the division algebras R, C, H, O.Comment: 11 page

Extremal Segments in Random Sequences

We investigate the probability for the largest segment in with total displacement $Q$ in an $N$-step random walk to have length $L$. Using analytical, exact enumeration, and Monte Carlo methods, we reveal the complex structure of the probability distribution in the large $N$ limit. In particular, the size of the longest loop has a distribution with a square-root singularity at $\ell\equiv L/N=1$, an essential singularity at $\ell=0$, and a discontinuous derivative at $\ell=1/2$.Comment: 3 pages, REVTEX 3.0, with multicol.sty, epsf.sty and EPS figures appended via uufiles. (Email in case of trouble.) CHANGES: Missing figure added to figures.uu MIT-CMT-KE-94-

Statistics of Largest Loops in a Random Walk

We report further findings on the size distribution of the largest neutral segments in a sequence of N randomly charged monomers [D. Ertas and Y. Kantor, Phys. Rev. E53, 846 (1996); cond-mat/9507005]. Upon mapping to one--dimensional random walks (RWs), this corresponds to finding the probability distribution for the size L of the largest segment that returns to its starting position in an N--step RW. We primarily focus on the large N, \ell = L/N << 1 limit, which exhibits an essential singularity. We establish analytical upper and lower bounds on the probability distribution, and numerically probe the distribution down to \ell \approx 0.04 (corresponding to probabilities as low as 10^{-15}) using a recursive Monte Carlo algorithm. We also investigate the possibility of singularities at \ell=1/k for integer k.Comment: 5 pages and 4 eps figures, requires RevTeX, epsf and multicol. Postscript file also available at http://cmtw.harvard.edu/~deniz/publications.htm

Randomly Charged Polymers, Random Walks, and Their Extremal Properties

Motivated by an investigation of ground state properties of randomly charged polymers, we discuss the size distribution of the largest Q-segments (segments with total charge Q) in such N-mers. Upon mapping the charge sequence to one--dimensional random walks (RWs), this corresponds to finding the probability for the largest segment with total displacement Q in an N-step RW to have length L. Using analytical, exact enumeration, and Monte Carlo methods, we reveal the complex structure of the probability distribution in the large N limit. In particular, the size of the longest neutral segment has a distribution with a square-root singularity at l=L/N=1, an essential singularity at l=0, and a discontinuous derivative at l=1/2. The behavior near l=1 is related to a another interesting RW problem which we call the "staircase problem". We also discuss the generalized problem for d-dimensional RWs.Comment: 33 pages, 19 Postscript figures, RevTe