Geometric inhomogeneous random graphs (GIRGs) are a class of random graphs where each vertex has an assigned weight coming from a power law distribution, and a spatial location according to a Poisson point process on the underlying geometric space. We consider GIRGs on the d-dimensional torus $\mathbb T^d := [-n^{1/d}/2,n^{1/d}/2)^d$ and perform percolation with edge-retention probability $\pi_n = \lambda n^{(3-\tau)/2}$ for some $\lambda > 0$. We prove that there exists a $\lambda_c$, such that for all $\lambda > \lambda_c$ a largest component unique in its order of magnitude exists within the highest $\ceil{an^{(\tau-3)/2}}$ weight vertices (the core) for some $a > 0$. Furthermore, we also prove novel results on the spatial distribution of the highest weight vertices. Starting from some fixed distance $r_0$, we sweep through the powers of 2 by defining $r_i := 2^ir_0$ and show that the amount of vertex pairs between $r_i$ and $r_{i+1}$ concentrates within its own expectation. We then consider the slightly supercritical regime, where we have already established the existence of a giant within the core. Extending the core to the entire graph, we establish that the largest component consists of mostly the emerging giant inside the core, together with the vertices reached outside the core, which we call its span. We then show a law of large numbers result on the size of the span of the giant component inside the core, where we show that for any $\varepsilon > 0$, we have that $\zeta^\lambda - \varepsilon \le \frac{\Span(\mathscr C_{(1)}^a)|}{\sqrt n}\leq \zeta^\lambda + \varepsilon$. ... Read more

We consider the problem of random walks moving around on a lattice Zd with an initial Poisson distribution of traps. We consider both static and moving traps. In the static case, we prove that the survival time has a decay of e−c t d /d +2 based on a heuristic argument. In the moving case we aim to prove the sub-exponential decay of the survival time in dimensions 1 and 2, as well as the exponential de- cay of survival in dimensions 3 and higher. We achieve the former by first expressing the survival probability in the range of a random walk and by showing that the asymptotic behavior of said range behaves in a sub-exponential and exponential way for dimensions 1/2 and ≥ 3 respectively. Further- more, we also show an upper bound for the survival time of the form lim supt →∞ 1 t log P(T ≥ t ) < 0. Following this we look at the situation where traps decay as ||x|| → ∞. Meaning less traps will be distributed further away from the origin. We show that in the static case, if the decay rate satisfies the condition px ≤ p(||x||) where p(r ) is non-increasing and r p(r ) is integrable and convergent, that the random walk will be transient, meaning that there will be a strictly positive chance of survival. Lastly, we then show that for the dynamically moving traps case, if the decay rate is "fast enough", meaning that if the Poisson parameter of the distribution of the traps ρ(x) is of the form 1/||x||2+α where α > d − 2, that there will also be a strictly positive probability of survival. ... Read more