Transitions for exceptional times in dynamical first-passage percolation

In first-passage percolation (FPP), we let(tau(v))be i.i.d. nonnegative weights on thevertices of a graph and study the weight of the minimal path between distant vertices. If Fi s the distribution function of tau v, there are different regimes: if F(0) is small,this weight typically grows like a linear function of the distance, and when F(0) islarge, the weight is typically of order one. In between these is the critical regime inwhich the weight can diverge, but does so sublinearly. We study a dynamical version ofcritical FPP on the triangular lattice where vertices resample their weights according toindependent rate-one Poisson processes. We prove that if n-expressionry sumexpressiontion (F-1)(1/2+1/2(k))=infinity,then a.s. there are exceptional times at which the weight grows atypically, but if n-expressionry sumexpressiontion k(7/8)F(-1)(1/2+1/2(k))<infinity, then a.s. there are no such times. Furthermore, in theformer case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider rangeof dynamical behavior than one sees in subcritical (usual) FPP.