arXiv subjects

Julian Ransford

Publications and source records attributed to Julian Ransford.

3 recordsLinked to original sources

arXiv2024

Limit shape formulas for a generalized Seppäläinen-Johansson model

We consider a simplified model of first-passage percolation, involving two families of i.i.d. random variables $\{ξ_{ij}\}$ and $\{η_{ij}\}$ corresponding to the weights of the horizontal and vertical edges respectively. We obtain an explicit formula for the limiting shape of the first-passage distance expressed in terms of the corresponding limit shapes of the two sets of weights for the Seppäläinen--Johansson model. We also study the limiting fluctuations of this model when at least one of the sets of weights is Bernoulli distributed.

math.PR↗

arXiv2023

Universality of directed polymers in the intermediate disorder regime

In 2018, Krishnan and Quastel showed that the fluctuations of Seppäläinen's log-gamma polymer converge in law to the Tracy--Widom GUE distribution in the intermediate disorder regime, which corresponds to taking the inverse temperature $β$ to depend on the length of the polymer $2n$, with $β=n^{-α}$ for some $α<1/4$. They also conjectured that this should hold for directed polymers with general i.i.d weights. We prove that their conjecture is true for any $1/5<α<1/4$.

math.PR↗

arXiv2019

A Gleason-Kahane-Żelazko theorem for the Dirichlet space

We show that every linear functional on the Dirichlet space that is non-zero on nowhere-vanishing functions is necessarily a multiple of a point evaluation. Continuity of the functional is not assumed. As an application, we obtain a characterization of weighted composition operators on the Dirichlet space as being exactly those linear maps that send nowhere-vanishing functions to nowhere-vanishing functions. We also investigate possible extensions to weighted Dirichlet spaces with superharmonic weights. As part of our investigation, we are led to determine which of these spaces contain functions that map the unit disk onto the whole complex plane.

math.FA↗