Ekaterina Toropova: Fluctuations and perturbations in spatial growth

Abstract: This thesis consists of two papers on spatial growth models. The first paper studies the travel time around a disc of large radius in first-passage percolation. The second concerns the noise sensitivity of the travel time in last-passage percolation.

In the first paper, we introduce and study a first-passage percolation setting in which direction plays a lesser role. More precisely, we consider the minimal travel time of a circuit around the origin that avoids a disc of radius \(n\). Under the same assumptions as in the classical Shape Theorem, we show that the circuit travel time grows linearly in \(n\) almost surely. Following the Newman--Piza approach, we also show that its variance is bounded below by a power of \(n\), matching the lower bound known for the point-to-point travel time in directions of curvature.

In the second paper, we are concerned with noise sensitivity of point-to-point travel time in geometric last-passage percolation. This is one of the first noise sensitivity results for a model in the Kardar--Parisi--Zhang (KPZ) universality class. In particular, we show that the travel time is noise sensitive under the perturbation acting on the Bernoulli encoding of geometric weights. To do this, we extend the Benjamini--Kalai--Schramm criterion to real-valued functions, bounding the correlation by a rescaled sum of squared influences. We then express the influence of a vertex in terms of the probability that it lies on the geodesic.