Biased random walk on critical Galton–Watson trees conditioned to survive

We consider the biased random walk on a critical Galton–Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain one-dimensional trapping models with slowly-varying tails. Indeed, in each of these two settings, we establish closely-related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs.     

Volume growth and heat kernel estimates for the continuum random tree

In this article, we prove global and local (point-wise) volume and heat kernel bounds for the continuum random tree. We demonstrate that there are almost–surely logarithmic global fluctuations and log–logarithmic local fluctuations in the volume of balls of radius r about the leading order polynomial term as r → 0. We also show that the on-diagonal part of the heat kernel exhibits corresponding global and local fluctuations as t → 0 almost–surely. Finally, we prove that this quenched (almost–sure) behaviour contrasts with the local annealed (averaged over all realisations of the tree) volume and heat kernel behaviour, which is smooth.      

Slow movement of a random walk on the range of a random walk in the presence of an external field

In this article, a localisation result is proved for the biased random walk on the range of a simple random walk in high dimensions ( $d\ge 5$ ). This demonstrates that, unlike in the supercritical percolation setting, a slowdown effect occurs as soon as a non-trivial bias is introduced. The proof applies a decomposition of the underlying simple random walk path at its cut-times to relate the associated biased random walk to a one-dimensional random walk in a random environment in Sinai’s regime. Via this approach, a corresponding aging result is also proved.     

Random Walk on the Range of Random Walk

We study the random walk X on the range of a simple random walk on ℤ ^d in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin.     

Heat kernel estimates for FIN processes associated with resistance forms

Quenched and annealed heat kernel estimates are established for Fontes–Isopi–Newman (FIN) processes on spaces equipped with a resistance form. These results are new even in the case of the one-dimensional FIN diffusion, and also apply to fractals such as the Sierpinski gasket and carpet.     

Heat kernel fluctuations for a resistance form with non-uniform volume growth

In this article, we consider the problem of estimating the heatkernel on measure-metric spaces equipped with a resistance form.Such spaces admit a corresponding resistance metric that reflectsthe conductivity properties of the set. In this situation, ithas been proved that when there is uniform polynomial volumegrowth with respect to the resistance metric the behaviour ofthe on-diagonal part of the heat kernel is completely determinedby this rate of volume growth. However, recent results haveshown that for certain random fractal sets, there are globaland local (point-wise) fluctuations in the volume as r 0 andso these uniform results do not apply. Motivated by these examples,we present global and local on-diagonal heat kernel estimateswhen the volume growth is not uniform, and demonstrate thatwhen the volume fluctuations are non-trivial, there will benon-trivial fluctuations of the same order (up to exponents)in the short-time heat kernel asymptotics. We also provide boundsfor the off-diagonal part of the heat kernel. These resultsapply to deterministic and random self-similar fractals, andmetric space dendrites (the topological analogues of graph trees).     

Local Limit Theorems for Sequences of Simple Random Walks on Graphs

In this article, local limit theorems for sequences of simple random walks on graphs are established. The results formulated are motivated by a variety of random graph models, and explanations are provided as to how they apply to supercritical percolation clusters, graph trees converging to the continuum random tree and the homogenisation problem for nested fractals. A subsequential local limit theorem for the simple random walks on generalised Sierpinski carpet graphs is also presented.      

Self-similarity and spectral asymptotics for the continuum random tree

We use the random self-similarity of the continuum random tree to show that it is homeomorphic to a post-critically finite self-similar fractal equipped with a random self-similar metric. As an application, we determine the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on the continuum random tree. We also obtain short time asymptotics for the trace of the heat semigroup and the annealed on-diagonal heat kernel associated with this Dirichlet form.