auto: 2026-05-06T16:35:22Z [skip ci]

content/essays/branch-based-local-capture-in-tree-balls/index.md

@@ -29,15 +29,15 @@ further-reading:
 
 ### Motivation
 
-The cops-and-robbers game is a pursuit-evasion game on a graph in which $k$ cops attempt to capture a single robber. The minimum $k$ for which the cops have a winning strategy is the *cop number* $c(G)$. *Meyniel's conjecture* (Frankl, 1987) asserts that $c(G) = O(\sqrt{n})$ for every connected graph on $n$ vertices, and remains the central open problem in the area. The current best upper bound is $c(G) \leq n / 2^{(1-o(1))\sqrt{\log n}}$ (Lu–Peng; Scott–Sudakov; Frieze–Krivelevich–Loh).
+The cops-and-robbers game is a [pursuit-evasion](https://en.wikipedia.org/wiki/Pursuit-evasion) game on a graph in which $k$ cops attempt to capture a single robber. The minimum $k$ for which the cops have a winning strategy is the *cop number* $c(G)$. [*Meyniel's conjecture*](https://en.wikipedia.org/wiki/Meyniel%27s_conjecture) (Frankl, 1987) asserts that $c(G) = O(\sqrt{n})$ for every connected graph on $n$ vertices, and remains the central open problem in the area. The current best upper bound is $c(G) \leq n / 2^{(1-o(1))\sqrt{\log n}}$ (Lu–Peng; Scott–Sudakov; Frieze–Krivelevich–Loh).
 
 A natural approach toward Meyniel's conjecture, in the spirit of Prałat–Wormald's resolution for random graphs and recent partial progress on expanders and high-girth graphs[@BradshawHosseiniMoharStacho; @HMG], is to combine random cop placement with a deterministic local chase. The bottleneck of this approach is the *local team-chase problem*: assuming several cops have been placed within a small distance $r$ of the robber, can they coordinate their gradient-descent moves to capture the robber within $O(r \log n)$ rounds?
 
-In high-girth regular graphs, where the local geometry is tree-like, the difficulty of this problem is structural: a single cop is always evaded (Aigner–Fromme), so capture requires multi-cop coordination, and the right coordination invariant is not obvious. In this paper, we work in the cleanest possible local geometry — a $d$-regular tree-ball — and identify the right invariant, prove that it works for one round, prove that it generalizes to $t$ rounds, and prove a matching barrier.
+In [high-girth](https://en.wikipedia.org/wiki/Girth_(graph_theory)) regular graphs, where the local geometry is tree-like, the difficulty of this problem is structural: a single cop is always evaded (Aigner–Fromme), so capture requires multi-cop coordination, and the right coordination invariant is not obvious. In this paper, we work in the cleanest possible local geometry — a $d$-regular tree-ball — and identify the right invariant, prove that it works for one round, prove that it generalizes to $t$ rounds, and prove a matching barrier.
 
 ### Setup
 
-Throughout, $G$ denotes a $d$-regular graph with $d \geq 3$. We fix a vertex $v_0 \in V(G)$ and a radius $R \geq 1$. The fundamental hypothesis is:
+Throughout, $G$ denotes a $d$-[regular graph](https://en.wikipedia.org/wiki/Regular_graph) with $d \geq 3$. We fix a vertex $v_0 \in V(G)$ and a radius $R \geq 1$. The fundamental hypothesis is:
 
 ::: {.annotation .annotation--static #def-tree-ball}
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@@ -57,7 +57,7 @@ In a tree-ball, $v_0$ has $d$ children (its neighbors), and every interior verte
 <span class="annotation-name">Geodesic cone</span>
 </div>
 <div class="annotation-body">
-For $v \in V(G)$ with $B_r(v)$ a tree-ball and $u \in N(v)$, the *geodesic cone* through $u$ at radius $r$ is
+For $v \in V(G)$ with $B_r(v)$ a tree-ball and $u \in N(v)$, the *[geodesic](https://en.wikipedia.org/wiki/Distance_(graph_theory)) cone* through $u$ at radius $r$ is
 
 $$C_u(v, r) := \{x \in B_r(v) \setminus \{v\} : \text{the unique geodesic from } x \text{ to } v \text{ has penultimate vertex } u\}.$$
 
@@ -118,7 +118,7 @@ The first main result is a probabilistic deep-load corollary.
 <span class="annotation-name">Deep-load occupancy</span>
 </div>
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-Let $G$ be $d$-regular with $B_r(v)$ a tree-ball, and let $X_1, \ldots, X_m$ be i.i.d. uniform samples from $B_r(v)$. Set
+Let $G$ be $d$-regular with $B_r(v)$ a tree-ball, and let $X_1, \ldots, X_m$ be [i.i.d.](https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables) uniform samples from $B_r(v)$. Set
 
 $$p_{d,r}^{\geq k} := \frac{|C_u^{\geq k}(v, r)|}{|B_r(v)|},$$
 
@@ -538,7 +538,7 @@ The crucial open question is whether one $t_0$-round epoch produces *permanent*
 - *Non-local data:* information that depends on the global graph structure beyond the local tree-ball, such as cycle structure at radius $> R$, or spectral data of the full graph.
 - *Asymmetric data:* information that breaks the rooted-tree symmetry, such as a fixed orientation, an external labeling, or a privileged subset of vertices not invariant under tree automorphisms.
 
-The most promising candidate among these is the second: *nonbacktracking-walk concentration* of cop mass on the full graph. If cops are placed by a global averaging that respects the nonbacktracking spectrum, the resulting cop distribution at time $t$ may concentrate on the robber's location at exponential rate $\rho(B)^{-1}$, where $B$ is the Hashimoto nonbacktracking transition matrix of the full graph. The local tree-ball view of such a placement is no longer order-$r$ for any fixed $r$; it inherits global spectral information that is invisible to any local invariant. This suggests that purely local tree-ball analysis is fundamentally insufficient.
+The most promising candidate among these is the second: *[nonbacktracking-walk](https://en.wikipedia.org/wiki/Non-backtracking_random_walk) concentration* of cop mass on the full graph. If cops are placed by a global averaging that respects the nonbacktracking spectrum, the resulting cop distribution at time $t$ may concentrate on the robber's location at exponential rate $\rho(B)^{-1}$, where $B$ is the Hashimoto nonbacktracking transition matrix of the full graph. The local tree-ball view of such a placement is no longer order-$r$ for any fixed $r$; it inherits global spectral information that is invisible to any local invariant. This suggests that purely local tree-ball analysis is fundamentally insufficient.
 
 ### Pivot C: Hierarchical packets
 
@@ -548,7 +548,7 @@ A naive cost analysis is discouraging: each scale costs $(d-1)^{2^j}$ cops by th
 
 ### Beyond the tree-ball model
 
-The deepest limitation of the present results is the tree-ball hypothesis. Real expanders — including Ramanujan graphs of girth $\Theta(\log n)$ — are tree-balls only up to radius $\Theta(\log n)$, but a chase argument typically needs to reason about radii $r \approx \log n / \gamma$ where $\gamma$ is the spectral gap. For $\gamma$ bounded below by a constant, these match up to constants; but for $\gamma$ small, the chase exits the tree-ball and must contend with cycles. In the cyclic regime, the geodesic cone $C_u(v, r)$ is no longer a clean object (vertices may have multiple shortest paths to $v$), and the entire branch decomposition becomes ill-defined.
+The deepest limitation of the present results is the tree-ball hypothesis. Real [expanders](https://en.wikipedia.org/wiki/Expander_graph) — including [Ramanujan graphs](https://en.wikipedia.org/wiki/Ramanujan_graph) of girth $\Theta(\log n)$ — are tree-balls only up to radius $\Theta(\log n)$, but a chase argument typically needs to reason about radii $r \approx \log n / \gamma$ where $\gamma$ is the [spectral gap](https://en.wikipedia.org/wiki/Spectral_gap). For $\gamma$ bounded below by a constant, these match up to constants; but for $\gamma$ small, the chase exits the tree-ball and must contend with cycles. In the cyclic regime, the geodesic cone $C_u(v, r)$ is no longer a clean object (vertices may have multiple shortest paths to $v$), and the entire branch decomposition becomes ill-defined.
 
 A full local team-chase theorem on expanders likely requires either (i) a generalization of the present analysis to graphs with bounded but nontrivial girth, where "tubes" become equivalence classes of approximately-tree-like geodesics, or (ii) a fundamentally different invariant — spectral, fence-based, or potential-theoretic — that does not rely on tree branch decomposition at all.