From 65cf33acfe0594e95f75c2840a9a020e5a4324f1 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 11 Mar 2025 13:52:57 -0300 Subject: Small wording tweaks in footnote. --- topology.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/topology.tex b/topology.tex index a55a5c4..cb95e01 100644 --- a/topology.tex +++ b/topology.tex @@ -365,15 +365,15 @@ problems that have a signal or spike, where it gives the overlap of a configuration with the hidden signal. Here $\textbf x_0$ is no signal, but a direction chosen uniformly at random and with no significance to the set of solutions. Here, if a feature of the action is present at some value $m$, it should be interpreted as indicating -that, with overwhelming probability, the feature is found in proximity to a typical -point in configuration space given by the overlap $m$. For instance, for $m$ +that, with overwhelming probability, typical configurations contributing to that feature have an overlap $m$ with a typical +point in configuration space. For instance, for $m$ sufficiently close to 1, $\mathcal S_\chi(m)$ is always negative, which is a result of the absence of any stationary points contributing to the Euler characteristic at those overlaps. Given a random height axis $\mathbf x_0$, the nearest point to $\mathbf x_0$ on the solution manifold will be the absolute -maximum of the height function, and therefore contribute to the Euler +maximum of the height function, and therefore will contribute to the Euler characteristic. Hence the region of negative action in -the vicinity of $m=1$ implies there is a typical distance between the +the vicinity of $m=1$ implies there is a typical minimum distance between the solution manifold and a randomly drawn point in configuration space, and that it is vanishingly unlikely to draw a point in configuration space uniformly at random and find it any closer to the solution manifold than this. -- cgit v1.2.3-70-g09d2