diff options
Diffstat (limited to 'topology.tex')
| -rw-r--r-- | topology.tex | 4 |
1 files changed, 3 insertions, 1 deletions
diff --git a/topology.tex b/topology.tex index 99c9a06..d3d90d4 100644 --- a/topology.tex +++ b/topology.tex @@ -409,7 +409,9 @@ $V_0=V_\text{\textsc{sat}}$ corresponding to the vanishing of the effective action at the $m=0$ solution, with $\mathcal S(0)=0$. For a generic covariance function $f$ it is not possible to write an explicit formula for $V_\text{\textsc{sat}}$, and we calculate it through a numeric -root-finding algorithm. +root-finding algorithm.\footnote{ +As a check of this calculation, the satisfiability threshold calculated here can be compared with that calculated using the zero-temperature limit of an equilibrium treatment of the cost function \eqref{eq:cost} made in Ref.~\cite{Urbani_2023_A} for the case where $f(q)=\frac12q^2$ and $\alpha=\frac14$. The authors estimate $V_\text{\textsc{sat}}\simeq1.871$, whereas this manuscript predicts $V_\text{\textsc{sat}}=1.867229\dots$, a seeming inconsistency. However, the author of Ref.~\cite{Urbani_2023_A} indicated in private correspondence that this difference could easily be explained by inaccuracy in the numeric \textsc{pde} treatment of the \textsc{frsb} equilibrium problem. Therefore, this manuscript is consistent with the previous work, but the agreement is not precise. +} When $V_0^2<V_\text{sh}^2$, the solution at $m=0$ is difficult to interpret, since the action takes a complex value. Such a result could arise from the breakdown |