Lower rounds for point-to-point wandering exponents in Euclidean first-passage percolation

In first-passage percolation models, the passage time T(0, L) from the orig in to a point L is expected to exhibit deviations of order \L\(chi) from it s mean, while minimizing paths are expected to exhibit fluctuations of orde r \L\(xi) away from the straight line segment (0L) over bar. Here, for Eucl idean models in dimension d, we establish the lower bounds xi greater than or equal to 1/(d + 1) and chi greater than or equal to (1 - (d - 1)xi)/2. C ombining this latter bound with the known upper bound xi less than or equal to 3/4 yields that chi greater than or equal to 1/8 for d = 2.