328,601 research outputs found
Dimension improvement in Dhar's refutation of the Eden conjecture
We consider the Eden model on the d-dimensional hypercubical unoriented
lattice , for large d. Initially, every lattice point is healthy, except the
origin which is infected. Then, each infected lattice point contaminates any of
its neighbours with rate 1. The Eden model is equivalent to first passage
percolation, with exponential passage times on edges. The Eden conjecture
states that the limit shape of the Eden model is a Euclidean ball. By putting
the computations of Dhar [Dha88] a little further with modern computers and
efficient implementation we obtain improved bounds for the speed of infection.
This shows that the Eden conjecture does not hold in dimension superior to 22
(the lower known dimension was 35)
Rationality of the Anomalous Dimensions in N=4 SYM theory
We reconsider the general constraints on the perturbative anomalous
dimensions in conformal invariant QFT and in particular in N=4 SYM with gauge
group SU(N_c). We show that all the perturbative corrections to the anomalous
dimension of a renormalized gauge invariant local operator can be written as
polynomials in its one loop anomalous dimension. In the N=4 SYM theory the
coefficients of these polynomials are rational functions of the number of
colours N_c.Comment: 20 pages, LaTe
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