« Minimal Cuts in the Z^D Lattice with Random Capacities
Minimal Cuts in the Z^D Lattice with Random Capacities
January 29, 2024, 2:00 PM - 3:00 PM
Location:
Hill Center, Room 005
Ron Peled, Tel-Aviv University
Endow the edges of the Z^D lattice with independent and identically distributed random capacities (e.g., uniformly distributed on [a,b] for some b>a>0). We wish to study the minimal cuts in the resulting network. Our focus is on the following setup: Consider the cube {-L,..., L}^D and the minimal cut separating the "upper" and "lower" halves of the boundary of the cube. How flat is this cut?
It is believed that the minimal cut is (power-law) rough in dimensions D=2,3 and flat in dimensions D>=6, and that it undergoes a roughening phase transition in dimension D=4 as the capacity distribution becomes more spread out. The case of dimension D=5 is more subtle. We discuss these predictions as well as rigorous results, and further mention rigorous results on related models of minimal surfaces in random environment.
The above combinatorial problem is further motivated by relations with first-passage percolation and disordered statistical physics models.
Based on joint works with Michal Bassan and Shoni Gilboa and with Barbara Dembin, Dor Elboim and Daniel Hadas.