Narrow Results

Rényi’s parking problem (or 1D sequential interval packing problem) dates back to 1958, when Rényi studied the following random process: Consider an interval $I$ of length $x$, and sequentially and randomly pack disjoint unit intervals in $I$ until the remaining space prevents placing any new segment. The expected value of the measure of the covered part of $I$ is $M(x)$, so that the ratio $M(x)/x$ is the expected filling density of the random process. Following recent work by Gargano et al. [4], we studied the discretized version of the above process by considering the packing of the 1D discrete lattice interval $\{1,2,\dots ,n+2k-1\}$ with disjoint blocks of $(k+1)$ integers but, as opposed to the mentioned [4] result, our exclusion process is symmetric, hence more natural. Furthermore, we were able to obtain useful recursion formulas for the expected number of $r$-gaps ($0\le r\le k$) between neighboring blocks. We also provided very fast converging series and extensive computer simulations for these expected numbers, so that the limiting filling density of the long line segment (as $n\to \infty$) is Rényi’s famous parking constant, $0.7475979203\dots .$

Let $(X,d)$ be a locally compact separable ultrametric space. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of all non-singleton balls $B$ of $X$, we consider the hierarchical Laplacian $L=L_C$. The operator $L$ acts in ${\mathcal{ℒ}}^2(X,m),$ is essentially self-adjoint and has a purely point spectrum. Choosing a sequence $\{𝜀(B)\}$ of i.i.d. random variables, we consider the perturbed function $C(B,\omega )$ and the perturbed hierarchical Laplacian $L^{\omega }=L_{C(\omega )}.$ Under certain conditions, the density of states $𝔪$ exists and it is a continuous function. We choose a point $\lambda$ such that $𝔪(\lambda )>0$ and build a sequence of point processes defined by the eigenvalues of $L^{\omega }$ located in the vicinity of $\lambda$. We show that this sequence converges in distribution to the homogeneous Poisson point process with intensity $𝔪(\lambda )$.