Universal fluctuations of first discoveries in competitive exploration

Random exploration is usually quantified by how fast new space is found, from
the range of a single walker to the territory collectively covered by many
walkers. In competitive exploration, first arrival secures an exclusive resource, as when foragers compete for food items or agents capture distributed targets. It is then no longer enough to know which sites have been discovered: one must determine, for each discovered site, which searcher reached it first. We introduce the discovery
share $X_n$, the fraction of the first $n$ collective discoveries secured by
a tagged searcher. For two identical competitors, exchange symmetry fixes
$\langle X_n\rangle=1/2$, but the central question is whether this equal
split emerges in each long exploration history or only on average, \emph{i.e.} whether early
competitive advantages are erased or persist. Here we show that the answer is
controlled by the spectral dimension $d_s$, defined by the large-time decay of the probability that a single searcher is at
its starting point after $t$ steps, $p_0(t)\sim t^{-d_s/2}$.
Across ordinary diffusion, long-range superdiffusion and subdiffusion induced
by crowding or memory, $d_s$ separates persistent randomness in
recurrent exploration $(d_s<2)$, anomalously slow non-Gaussian concentration
for $2\le d_s<3$, and Gaussian concentration, logarithmically corrected at
$d_s=3$, for $d_s\ge3$. For $d_s\ge2$, we derive exact asymptotic
variances, including prefactors, and the discovery scale on which competitive
imbalances are erased. Two-point correlations of first-discovery labels identify the memory mechanism behind these regimes.
The same phase structure persists under changes in geometry, competitor
heterogeneity, number of competitors and memory, revealing a general fluctuation
theory of first-arrival inequalities.