arXiv subjects
Publications and source records attributed to B. T. Graham.
A range of first-passage percolation type models are believed to demonstrate the related properties of sublinear variance and superdiffusivity. We show that directed last-passage percolation with Gaussian vertex weights has a sublinear variance property. We also consider other vertex weight distributions. Corresponding results are obtained for the ground state of the `directed polymers in a random environment' model.
Let $μ$ be the self-avoiding walk connective constant on $\ZZ^d$. We show that the asymptotic expansion for $β_c=1/μ$ in powers of $1/(2d)$ satisfies Borel type bounds. This supports the conjecture that the expansion is Borel summable.
The so-called diluted-random-cluster model may be viewed as a random-cluster representation of the Blume--Capel model. It has three parameters, a vertex parameter $a$, an edge parameter $p$, and a cluster weighting factor $q$. Stochastic comparisons of measures are developed for the `vertex marginal' when $q\in[1,2]$, and the `edge marginal' when $q\in[1,\oo)$. Taken in conjunction with arguments used earlier for the random-cluster model, these permit a rigorous study of part of the phase diagram of the Blume--Capel model.
The influence theorem for product measures on the discrete space {0,1}^N may be extended to probability measures with the property of monotonicity (which is equivalent to `strong positive-association'). Corresponding results are valid for probability measures on the cube [0,1]^N that are absolutely continuous with respect to Lebesgue measure. These results lead to a sharp-threshold theorem for measures of random-cluster type, and this may be applied to box-crossings in the two-dimensional random-cluster model.