An Indepth Look at Isoperimetry in the Cayley Graph Generated by All Transpositions Let $\Omega_n$ denote the symmetric/permutation group on $n$ objects.
Let $T_n \subseteq \Omega_n$ denote the set of transpositions.
Drop the $n$-subscripts.
Define the Cayley graph $G = (\Omega, E)$ by saying that $\sigma, \sigma' \in \Omega$ are adjacent if $\sigma^{-1} \sigma \in T$, ie they differ by a transposition.
One can use the Aldous spectral gap conjecture to show that the spectral gap of (the simple random walk on) this graph is $1/n$.
A standard result by multiple authors (Jerrum and Sinclair, to name two) then says that
$1/n^2 \lesssim \Phi_* \lesssim 1/n$ where $\Phi_*$ is the isoperimetric constant:
$$
\Phi(S) := \frac{|\partial S|}{|S|}
\quad\text{and}\quad
\Phi_* := \min_{|S| \le |\Omega|/2} \Phi(S),$$
where $\partial S$ is the edge boundary of the set $S$.
This is a worst-case bound.
I am interested in a better understanding of the isoperimetric profile $\Phi(S)$, not just for the worst-case $S$.
References and the like would be appreciated.

Consider this example.
Define $S$ by including every permutation independently with probability $\tfrac12$. (I just mean site $\tfrac12$-percolation by this.)
Fix any transposition $\tau \in T$.
Define
$$ \partial_\tau S
:=
\{ \sigma \tau \mid \sigma \in S \text{ and } \sigma \tau \notin S \}.
$$
One can pair up all the permutations: $(\sigma, \sigma \tau)$ where $\sigma$ ranges over a set of size $\tfrac12 |\Omega|$.
This outlook shows that
$$
|\partial_\tau S|
\sim
\textrm{Bin}(|\Omega|, \tfrac14).
$$
Indeed, for every point in $S$ there is a $\tfrac12$ chance that its pair is not in $S$.
Also, $|S| \sim \textrm{Bin}(|\Omega|, \tfrac12)$.
Thus
$ \Phi(S) \approx \tfrac12. $
So if $S$ is drawn in this sense then typically it has isoperimetric expansion roughly $\tfrac12$.
This is much better than the worst-case, which is at least as bad as $1/n$.

Ideally I would like to determine some characterisation of the set of 'expanding' sets, ie ones with $\Phi(S) \asymp 1$.
More formally, define $\mathcal P_c := \{ S \subseteq \Omega \mid \Phi(S) \ge c \}$ for $c \ge 0$.
Then $\mathcal P_0$ is just the power set of $\Omega$.
I am after some characterisation of $\mathcal P_c$, or some large subset of it, for $c > 0$ fixed and $n \to \infty$.
 A: This reply is a bit too long for a comment, but it responds mainly to the comment from the OP who wrote that "it surely must be the case that $\Phi(S_t)$ is typically of order 1".   The mixing time bound using evolving sets  need not be sharp, and there are results (see [2]) that support  Benoît Kloeckner's intuition that the evolving sets are typically "round", i.e., provide sparse cuts.
The full spectrum (not just the spectral gap) of random transpositions has been known long before the Aldous conjecture, see [3].
[1] Morris, Ben, and Yuval Peres. "Evolving sets, mixing and heat kernel bounds." Probability Theory and Related Fields 133, no. 2 (2005): 245-266.
[2]  Andersen, Reid, and Yuval Peres. "Finding sparse cuts locally using evolving sets." In Proceedings of the forty-first annual ACM symposium on Theory of computing, pp. 235-244. 2009.
[3] Diaconis, Persi, and Mehrdad Shahshahani. "Generating a random permutation with random transpositions." Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 57, no. 2 (1981): 159-179.
