The First Betti Number Of Continuous Loop Groups

Let G be a connected compact Lie group with a unit element e, L_e(G)the corresponding continuous loop group pinned at e andμ_e the pinned Wiener measure on L_e(G). Then the first Betti number b₁ for H₁-bundle over L_e(G)is infinite provided G is of nontrivial center.Moreover, for RWRE on 3-regular planar graphs with uniformly elliptic product random environments, the 0-1 law for it tending to infinity along a fixed direction or its inverse direction holds.