Dimension improvement in Dhar's refutation of the Eden conjecture

Highlights: • We show that the Eden conjecture does not hold in dimension superior to 22. • Dhar's computations (Dhar, 1988) are pushed a little further with modern computers. • Bounds for the speed of infection along the axis are improved. • This result is combined with a bound for the speed of infection along the diagonal. - Abstract: 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 pushing the computations of Dhar [5] 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 lowest known dimension was 35).