Year: 2026
Type: preprint
Abstract: We study the noise sensitivity of Boolean functions on the symmetric group $S_n$, where noise is induced by running a continuous-time Markov chain on $S_n$. We investigate the noise sensitivity of several fundamental functions, including the parity function and analogues of the dictator function. We focus, in particular, on the case where the underlying Markov chain is an interchange process on the complete graph. We establish comparison results between noise sensitivity with respect to the interchange process on the complete graph and the weighted interchange process on any connected graph. We also show that the indicator function of long cycles is noise sensitive under the interchange process on each of these graphs. Finally, using the fact that the interchange process on the complete graph $K_n$ is the continuous-time random walk generated by all transpositions, we prove that noise sensitivity is preserved when the noise source is changed to the continuous-time random walk generated by all $s$-cycles, where $s$ is even and $2
Source: arXiv (Cornell University)
Authors: Amir, Gideon, Ghosh, Subhajit
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166675626