Sharp Phase Transitions for k-Fold Coverage Using Morse Theory

TL;DR

This work develops a Morse-theoretic framework for analyzing random $k$-fold coverage on a torus, linking the topology of the $k$-coverage set to the critical points of the $k$-NN distance function. It establishes a sharp phase transition for $k$-coverage at the threshold $\Lambda=\log n+(d+k-2)\log\log n$, with a concurrent Poisson-process description of the last uncovered regions in the critical window. The analysis extends Morse theory to the non-smooth $k$-NN distance and derives detailed counts and variance for critical points of all indices, yielding topological consequences such as the Euler characteristic behavior and implications for homological connectivity. Collectively, the results provide precise probabilistic and topological characterizations of coverage in dense random geometric settings, with a novel methodological bridge between stochastic geometry and Morse theory applicable to manifold settings.

Abstract

We introduce a novel approach for studying random k-coverage, using Morse theory for the k-nearest neighbor (k-NN) distance function. We prove a sharp phase transition for the number of critical points of the k-NN distance function, from which we conclude a phase transition for k-coverage. In addition, in the critical window our new framework enables us to prove a Poisson process approximation (in both location and size) for the last uncovered regions.

Figures (5)

• Figure 1: Homology. Left: The 2-dimensional sphere $\mathbb{S}^2$ has a single connected component ($0$-cycle), enclosing an "air pocket" ($2$-cycle). Hence, $\beta_0(\mathbb{S}^2)=\beta_2(\mathbb{S}^2)=1$, and $\beta_i(\mathbb{S}^2)=0$ for all $i\neq 0,2$. Center: The torus ${\mathbb{T}}^2$ has one connected component, two independent $1$-cycles (dashed lines) and a single $2$-cycle. Hence, $\beta_0({\mathbb{T}}^2)=1,\beta_1({\mathbb{T}}^2)=2,\beta_2({\mathbb{T}}^2)=1$. Right: A planar graph $G$ on $12$ vertices, with three connected components and a single $1$-cycle. Hence, $\beta_0(G)=3$ and $\beta_1(G)=1$.
• Figure 2: Critical points of $d_{{\cal{P}}}^{(k)}$ in ${\mathbb{R}}^2$, for $k=2$. Left: The set ${\cal{X}}_1=\{x_1,x_2,x_3\}$ induces a critical point $c$ of index $\mu = 2$, since the interior of ${\cal{B}}({\cal{X}}_1)$ contains exactly a single point $y_1$, and $\sigma({\cal{X}}_1)$ (dashed triangle) includes $c$. The shaded purple region is $B_r^{(2)}({\cal{P}})$. Center: the set ${\cal{X}}_2=\{x_4,x_5,x_6\}$ does not induce a critical point, since $c\not\in\sigma({\cal{X}}_2)$. Right: the set ${\cal{X}}_3=\{x_7,x_8,x_9\}$ does not induce a critical point, since the interior of ${\cal{B}}({\cal{X}}_3)$ includes more than one point.
• Figure 3: A configuration of two critical points of index $\mu=2$ in ${\mathbb{R}}^2$ for $k=4$. The set ${\cal{X}}=\{x_1,\ldots,x_5\}$ induces two critical points, generated by ${\cal{X}}_1=\{x_1,x_2,x_3\}$ and ${\cal{X}}_2=\{x_3,x_4,x_5\}$. Here $k_1=1$, $k_2=1$, $k_{12}=1$, $m_1=1$ and $m_2=1$. In addition, $j=1$ since ${\cal{X}}_1$ and ${\cal{X}}_2$ share the point $x_3$. The volume $V_1({\cal{X}})$ is green, $V_2({\cal{X}})$ is blue, and $V_{12}({\cal{X}})$ is yellow.
• Figure 4: $I_j^{(2)}$ configurations in ${\mathbb{R}}^2$ for $k=2$ and $j=2$. The critical points $c_1$ and $c_2$ of index $\mu=2$ are induced by ${\cal{X}}_1=\{x_1,x_3,x_4\}$ and ${\cal{X}}_2=\{x_2,x_3,x_4\}$, respectively. Since $j=|{\boldsymbol{x}}_1\cap{\boldsymbol{x}}_2|=2$ and $\delta_j<\epsilon_j$, the centers $c_1$ and $c_2$ lie on the same side of the dashed line connecting $x_3$ and $x_4$, resulting in ${\cal{B}}({\boldsymbol{x}}_1)$ (enclosed by the green circle) including more than half the sphere $S({\boldsymbol{x}}_2)$ (blue circle). Thus, $c_2$ critical implies that at least one point of ${\boldsymbol{x}}_2$ must lie inside ${\cal{B}}({\boldsymbol{x}}_1)$ (the point $x_1$).
• Figure 5: $I_j^{(2)}$ for $m_1=0$ in ${\mathbb{R}}^3$. (a) The circle ($1$-sphere) $S_3$ (black dashed line) splits the critical sphere of $c_2$ (in blue) to two hemispheres, one of them is $\hat{S}_3$ (brown dashed line). $c_2$ critical implies than one of the points of the associated configuration must lie in the red region defined by the part of $\hat{S}_3$ not contained in ${\cal{B}}({\boldsymbol{x}}_1)$ (the green sphere). (b) Side view of the critical configuration.

Theorems & Definitions (35)

• Theorem 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Theorem 3.1
• Theorem 3.2
• Corollary 3.3
• Theorem 3.4
• Proposition 3.5
• Corollary 3.6