abstract:
The lace
expansion has been a powerful tool to investigate mean-field critical
behavior for stochastic-geometrical models, such as self-avoiding walk and
percolation, above the upper-critical dimension. We recently developed the
lace expansion for the Ising model. Applying this expansion to the
ferromagnetic case, we can prove that the model exhibits the mean-field
behavior if the dimension d is greater than 4 and the coordination number N
(= 2d for the nearest-neighbor model, for example) is sufficiently large.
The main point of this approach is that we do not have to require the
reflection positivity of the spin-spin coupling, which has long been an
essential ingredient for the proof of the mean-field behavior.
In the talk, I will briefly explain 1) the derivation of the lace expansion
for the Ising model, 2) the implication assuming its convergence, and 3) the
reason why the expansion converges when d>4 and N>>1.
abstract:
Edge-reinforced random walk on any
finite graph has the same distribution as a random walk in a random
environment, where the environment is given by random, but time-independent
weights on the edges.
This result is well-known for finite graphs. In the talk we will present the
result for arbitrary infinite graphs. The proof relies on a comparison with
Polya urns.
This comparison yields some bounds for the random environment.
The talk is based on a joint paper with Franz Merkl.