& (\Ll_{0} f)(\eta) = \frac{1}{2} N^{\alpha} \sum\limits_{x=1}^{N-1} (f(\eta^{x, x+1}) - f(\eta)) + \frac{1}{2} N^{\alpha} \sum\limits_{x=1}^{N} \left[ (f(\eta^{x,+}) - f(\eta)) + (f(\eta^{x,-}) - f(\eta)) \right].
\end{align*}

Here as usual $\eta^{x, \pm}$ denotes the configuration obtained from $\eta$ by changing the color $\phi_x$ of the particle at site $x$ to $\phi_x \pm 1$ (note that the colors $a_i$ do not evolve in time here). We will then transfer the result to the biased process by estimating its Radon-Nikodym derivative.

Let $\Pp_{0}^{N}$ be the law of the unbiased process with rates $1$ described above (with an arbitrary initial distribution). With the notation from Lemma <ref>, we have for any $t \geq 0$ and $\delta > 0$

\[
\limsup\limits_{l \to \infty} \limsup\limits_{N \to \infty} N^{-\gamma} \log \Pp_{0}^{N} \left( \int\limits_{0}^{t} U^{N}_{l}(\eta_s) \, ds  > \delta \right) = - \infty.
\]

Let us write $\Ll_{0} = \Ll + \Ll_{c}$, where $\Ll$ is the first term in the definition of $\Ll_{0}$ and $\Ll_{c}$ is the second term. The dynamics induced by $\Ll$ and by $\Ll_{c}$ is reversible with respect to $\mu$, so the Dirichlet forms associated respectively to $\Ll_{c}$ and $\Ll_{0}$ can be written as

\begin{align*}
& D^{N}_{c}(f) = \frac{1}{4} N^{\alpha} \int \sum\limits_{x=1}^{N} \left[ \left( \sqrt{f(\eta^{x,+})} - \sqrt{f(\eta)} \right)^2 + \left( \sqrt{f(\eta^{x,-})} - \sqrt{f(\eta)} \right)^2 \right]\, d\mu(\eta), \\
& D^{N}_{0}(f) = D^{N}(f) + D^{N}_{c}(f).
\end{align*}

By repeating the argument from the proof of Lemma <ref> with the generator $\Ll_0$ instead of $\Ll$ we obtain that it is enough to prove that for any $c > 0$

\[
\limsup\limits_{l \to \infty} \limsup\limits_{N \to \infty} \sup\limits_{f} \left\{ \int c U^{N}_{l}(\eta)f(\eta) \, d\mu(\eta) - N^{-\gamma} D^{N}_{0}(f) \right\} \leq 0,
\]

where the supremum is taken over all densities with respect to $\mu$.

Now observe that since $D^{N}_{c}(f) \geq 0$ for any nonnegative $f$, it is in fact enough to prove the statement above with $D^{N}_{0}(f)$ replaced by $D^{N}(f)$. Thus we have eliminated color evolution and the conclusion follows as in the proof of Lemma <ref>.

We can now prove the superexponential estimate for the biased process.

Recall that $f_x(\eta) = L_{x}(\eta) v_{x-1}(\eta)$. Since we can uniformly approximate $v(0,x,\phi)$ by finite sums of terms which are product in $x$ and $\phi$, by using the triangle inequality we can without loss of generality assume that $v_{x}(\eta) = \psi(x)g(\phi_x)$ for some continuous functions $\psi : [0,1] \to \R, g : [0,1] \to [-1,1]$. Applying Lemma <ref> with $w(x, \phi)= v(0,x,\phi)$, $a_i = L_i$ and $h_x = L_x g(\phi_{x-1})$ provides us with the superexponential estimate for the process $\Pp_{0}^{N}$. To transfer the estimate to the biased process $\widetilde{\Pp}^{N}$ we will need to estimate the Radon-Nikodym derivative of the two processes.

If $\Pp$ is a Markov process with jump rates $\lambda(x)p(x,y)$ and $\widetilde{\Pp}$ is another process on the same state space with rates $\widetilde{\lambda}(x) \widetilde{p}(x,y)$, the Radon-Nikodym derivative up to time $t$ is given by (see, e.g., <cit.>)

\begin{equation}\label{eq:radon-nikodym-sum-rates}
\frac{\dd \widetilde{\Pp}}{\dd \Pp} (t)  = \exp \left\{ - \int\limits_{0}^{t} \left( \widetilde{\lambda}(X_{s}) - \lambda(X_{s}) \right) ds + \sum\limits_{s \leq t}\log \frac{\widetilde{\lambda}(X_{s-})\widetilde{p}(X_{s-},X_{s})}{\lambda(X_{s-})p(X_{s-},X_{s})}\right\},
\end{equation}

where the sum is over jump times $s \leq t$.

Let us look at $\frac{\dd \widetilde{\Pp}^{N}}{\dd \Pp_{0}^{N}}$. By the form (<ref>) of the generator of $\widetilde{\Pp}^{N}$ the sum of outgoing rates for any $\eta$ is equal to

\[
\frac{1}{2}N^{\alpha} \left(\sum\limits_{x=1}^{N-1} \left[ 1 + \varepsilon \left(v_{x}(\eta) - v_{x+1}(\eta) \right) \right] + \sum\limits_{x=1}^{N} \left[ 1 + \varepsilon r_{x}(\eta) \right]  + \sum\limits_{x=1}^{N} \left[ 1 - \varepsilon r_{x}(\eta) \right]\right).
\]

Since the sum of $\varepsilon(v_{x} - v_{x+1})$ telescopes, the rates $v_x$ are $0$ at the boundaries $x=1, N$ and rates $r_x$ for the color change cancel out, the intensities $\widetilde{\lambda}$ and $\lambda$ cancel out as well. The Radon-Nikodym derivative takes the form

\begin{align}\label{eq:radon-nikodym-sum-jumps}
\frac{\dd \widetilde{\Pp}^{N}}{\dd \Pp_{0}^{N}}(t)  = &  \exp \Big\{ \sum\limits_{s \leq t} \log \left( 1 + \varepsilon \left[v(x_{j_{s}}, \phi_{x_{j_{s}}}(\eta_{s})) - v(x_{j_{s}}+1, \phi_{x_{j_{s}}+1}(\eta_{s}))\right] \right) + \\
& + \sum\limits_{s_{+} \leq t} \log \left( 1 + \varepsilon \left[r(x_{j_{s_{+}}}, \phi_{x_{j_{s_{+}}}}(\eta_{s_{+}}))\right] \right) + \sum\limits_{s_{-} \leq t} \log \left( 1 - \varepsilon \left[r(x_{j_{s_{-}}}, \phi_{x_{j_{s_{-}}}}(\eta_{s_{-}}))\right] \right) \Big\},\nonumber
\end{align}

where $j_{s}$ is the label of the particle which makes a swap at time $s$ and $j_{s_{\pm}}$ is the label of the particle that changes its color by $\pm 1$ at time $s_{\pm}$.

To simplify this formula we will use the fact that empirical currents across edges can be approximated by their averages, modulo a small martingale. More precisely, let us denote for simplicity

\[
(\nabla_{x} v)(\eta) = v_x (\eta) - v_{x+1} (\eta).
\]

We will sometimes use this notation with $x=N$, in which case we assume $(\nabla_{x} v)(\eta) = 0$. For brevity of notation whenever sums involving both $r_x$ and $-r_x$ appear, we will write them as one term with a $\pm$ sign, that is, with $\sum\limits_{x}(1 \pm \varepsilon r_x)$ serving as a shorthand for $\sum\limits_{x}(1 + \varepsilon r_x) + \sum\limits_{x}(1 - \varepsilon r_x)$ and so on.

We introduce the following extension of the dynamics under $\Pp_{0}^{N}$ – for any functions $h(x,\eta)$, $h^{\pm}(x,\eta)$, $x \in \{1, \ldots, N\}$ consider the extended state space $E'$, consisting of pairs $(\eta, J)$, $J \in \mathbb{R}$, and the generator $\Ll'$ acting by

\begin{align*}
(\Ll' f)(\eta, J) = & \frac{1}{2} N^{\alpha} \Big[ \sum\limits_{x=1}^{N-1} \left( f(\eta^{x,x+1}, J + h(x, \eta)) - f(\eta, J) \right) +\\
& + \sum\limits_{x=1}^{N} \left( f(\eta^{x,+}, J + h^{+}(x, \eta)) - f(\eta, J) \right) + \sum\limits_{x=1}^{N} \left( f(\eta^{x,-}, J + h^{-}(x, \eta)) - f(\eta, J) \right) \Big].
\end{align*}

In other words, in the evolution of the extended configuration $(\eta_t, J_t)$ each time the process makes a jump, $J_t$ is increased by $h(x, \eta_t)$, $h^{+}(x, \eta_t)$ or $h^{-}(x, \eta_t)$, depending on the type of the jump (swap or color change). Now if we take

\begin{align*}
& h(x,\eta) = \log \left[ 1 + \varepsilon (\nabla_{x} v)(\eta)\right], \\
& h^{\pm}(x,\eta) = \log \left[ 1 \pm \varepsilon r_{x}(\eta)\right],
\end{align*}

we see that $J_t$ is simply equal to the sum over jumps appearing in the exponent in (<ref>). Thus to bound the Radon-Nikodym derivative we only need to bound $J_t$.

This is done by use of an exponential martingale – for any $\lambda > 0$ the following process

\[
Z_{t} = \exp\left\{ \lambda J_{t}- \int\limits_{0}^{t} e^{-\lambda J_{s}} \Ll' e^{\lambda J_{s}} \, ds \right\}
\]

is a local martingale with respect to $\Pp_{0}^{N}$. We will actually only need to consider $\lambda = 2$. Writing out the action of $\Ll'$ on the function $g(\eta, J) = e^{2 J}$ we obtain

\[
Z_{t} = \exp\left\{ 2 J_{t}- \frac{1}{2} N^{\alpha} \int\limits_{0}^{t} \sum\limits_{x=1}^{N} \left[ \left( e^{2 \log(1 + \varepsilon (\nabla_{x} v)(\eta_{s}))} - 1\right) +  \left( e^{2 \log(1 \pm \varepsilon r_x (\eta_{s})} - 1 \right) \right]  ds \right\}.
\]

Now we have

\begin{align*}
& e^{2 \log(1 + \varepsilon (\nabla_{x} v)(\eta_{s}))} - 1 = (1 + \varepsilon (\nabla_{x} v)(\eta_{s}))^2 - 1 = 2 \varepsilon (\nabla_{x} v)(\eta_{s}) + \varepsilon^2 \left[(\nabla_{x} v)(\eta_{s})\right]^2, \\
& e^{2 \log(1 \pm \varepsilon r_{x}(\eta_{s}))} - 1 = (1 \pm \varepsilon r_{x}(\eta_{s}))^2 - 1 = \pm 2 \varepsilon r_{x}(\eta_{s}) + \varepsilon^2 r_{x}(\eta_{s})^2.
\end{align*}

The sum of terms linear in $\varepsilon$ vanishes – the rates $r$ for $\pm 1$ color change have opposite sign and the sum involving $\nabla_{x}v$ telescopes. Recalling that $\varepsilon = N^{1-\alpha}$ and $\gamma = 3 - \alpha$, so $N^{\alpha+1}\varepsilon^2 = N^\gamma$, we can then write

\[
Z_{t} = \exp\left\{ 2 J_{t} - \frac{1}{2} N^{\gamma} \int\limits_{0}^{t} \frac{1}{N}\sum\limits_{x=1}^{N} \left[ (\nabla_{x} v)(\eta_{s})^2  + r_x(\eta_{s})^2\right] \, ds \right\}.
\]

Since the rates $v$ and $r$ are bounded, we have $Z_t = e^{ 2 J_t - N^{\gamma} X_t}$, where $|X_t| \leq C$ for some constant $C > 0$ depending only on $v$, $r$ and $T$. In particular we get

\[
\E e^{2 J_t} = \E \left( e^{2 J_t - N^{\gamma} X_t} e^{N^{\gamma}X_t} \right) = \E \left( Z_t e^{N^{\gamma}X_t} \right) \leq e^{C N^{\gamma}} \E Z_t.
\]

Since $Z_t$ is a local martingale bounded from below, it is a supermartingale, so we have $\E Z_t \leq \E Z_0 = 1$ and thus

\begin{equation}\label{eq:exponential-zt}
\E e^{2 J_t} \leq e^{C N^\gamma}.
\end{equation}

Now we can transfer the superexponential bound of Lemma <ref> from $\Pp_{0}^{N}$ to $\widetilde{\Pp}^{N}$. Let $\Oo_{N,l}$ be the event from the statement of the lemma and let us write simply $\frac{\dd \widetilde{\Pp}^{N}}{\dd \Pp_{0}^{N}} = \frac{\dd \widetilde{\Pp}^{N}}{\dd \Pp_{0}^{N}}(T)$. Denoting by $\widetilde{\E}$ the expectation with respect to $\widetilde{\Pp}^{N}$ we have

\[
\widetilde{\Pp}^{N}\left( \Oo_{N,l} \right) = \widetilde{\E} \left( \id_{\Oo_{N,l}} \right) = \E \left( \frac{\dd \widetilde{\Pp}^{N}}{\dd \Pp_{0}^{N}} \id_{\Oo_{N,l}} \right).
\]

Applying the Cauchy-Schwarz inequality gives

\[
\widetilde{\Pp}^{N}\left( \Oo_{N,l} \right) \leq \left[ \E \left(\frac{\dd \widetilde{\Pp}^{N}}{\dd \Pp_{0}^{N}}\right)^2 \right]^{1/2} \cdot \Pp_{0}^{N}\left(\Oo_{N,l}\right)^{1/2}.
\]

Recalling that $\frac{\dd \widetilde{\Pp}^{N}}{\dd \Pp_{0}^{N}} = e^{J_T}$ and applying the bound (<ref>) we obtain

\[
\widetilde{\Pp}^{N}\left( \Oo_{N,l} \right) \leq e^{c N^{\gamma}} \Pp_{0}^{N}\left(\Oo_{N,l}\right)^{1/2}
\]

with $c = \frac{C}{2}$. Thus

\[
\limsup\limits_{N \to \infty} N^{-\gamma} \log \widetilde{\Pp}^{N}\left( \Oo_{N,l} \right) \leq  c + \frac{1}{2}\limsup\limits_{N \to \infty} N^{-\gamma} \log \Pp_{0}^{N}\left( \Oo_{N,l} \right)
\]

and taking $\limsup$ as $l \to \infty$ together with an application of Lemma <ref> finishes the proof.

Large deviation lower bound

In this section we prove the large deviation lower bound of Theorem <ref>. Let us assume that the permuton process $X$ satisfies equations <ref>. Since we already know how to construct a biased interchange process that will typically display the behavior of $X$, to bound the probability that the trajectory of a random particle in the interchange process is close in distribution to $X$ we only need to compare the unbiased process with the biased one by means of calculating their Radon-Nikodym derivative.

Since these two processes have different configuration spaces, for convenience we introduce the unbiased interchange process with colors, which has the same configuration space as the biased process associated to <ref> and the generator $\Ll^{u}$ obtained by putting all velocities $v$ to $0$

\begin{align}\label{eq:unbiased-with-colors}
& (\Ll^{u}_{t} f)(\eta) = \frac{1}{2} N^{\alpha} \sum\limits_{y=1}^{N-1} (f(\eta^{y, y+1}) - f(\eta))  + \frac{1}{2} N^{\alpha} \sum\limits_{x=1}^{N} \left[ 1 \pm \varepsilon r(t, x, \phi_{x}(\eta)) \right] (f(\eta^{x,\pm}) - f(\eta)).
\end{align}

Since here the colors do not influence the dynamics of swaps, the corresponding permutation process $X^{\eta^{N}}$ will be the same as for the ordinary unbiased interchange process (and we will never be interested in the distribution of $\Phi^{\eta^{N}}$ for the unbiased process with colors).

Let us start by deriving the formula for the Radon-Nikodym derivative of the unbiased process with colors with respect to the biased one. Recall that $v_x (s, \eta_{s}) = v(s,x,\phi_{x}(\eta_s))$ denotes the velocity at time $s$ of the particle at site $x$. Let $\Pp^{N}_{u}$ denote the law of the unbiased process with colors. We will prove the following statement

We have

\[
\frac{\dd \Pp_{u}^{N}}{\dd \widetilde{\Pp}^{N}}(T) = \exp \Bigg\{ - \frac{1}{2} N^{\gamma} \Bigg[ \int\limits_{0}^{T} \frac{1}{N} \sum\limits_{x=1}^{N} v_x (s, \eta_{s})^2 \, ds + o(1) \Bigg]\Bigg\},
\]

where the $o(1)$ term goes to $0$ in probability as $N \to \infty$.

The calculation is similar as in the proof of Lemma <ref>, with the difference that we are using generator $\widetilde{\Ll}$ instead of $\Ll_0$. By the analog of formula (<ref>) for time-inhomogeneous processes we have

\[
\frac{\dd \Pp_{u}^{N}}{\dd \widetilde{\Pp}^{N}}(t)  = \exp \left\{ -\sum\limits_{s \leq t} \log \left( 1 + \varepsilon \left[v(s,x_{j_{s}}, \phi_{x_{j_{s}}}(\eta_{s})) - v(s,x_{j_{s}}+1, \phi_{x_{j_{s}}+1}(\eta_{s}))\right] \right)  \right\},
\]

where the sum is over jump times $s \leq t$.

Denoting the sum in the exponent by $J_{t}$, we obtain by (<ref>) (by considering as before the generator $\widetilde{\Ll}$ acting on an extended configuration space) that

\[
J_{t} = M_{t} + \frac{1}{2} N^{\alpha}\sum\limits_{x=1}^{N-1} \int\limits_{0}^{t} \left[ 1 + \varepsilon (\nabla_{x} v)(s, \eta_{s})\right] \log \left[ 1 + \varepsilon (\nabla_{x} v)(s, \eta_{s})\right] ds,
\]

where $M_{t}$ is a local martingale with respect to $\widetilde{\Pp}^{N}$. Expanding all terms up to order $\varepsilon^2$ allows us to write

\[
\frac{\dd \Pp_{u}^{N}}{\dd \widetilde{\Pp}^{N}}(t)  = \exp \left\{- M_{t} -  \frac{1}{2} N^{\alpha} \int\limits_{0}^{t} \sum\limits_{x=1}^{N-1} \left[ \varepsilon (\nabla_{x} v)(s,\eta_{s}) + \frac{\varepsilon^2}{2} \left[ (\nabla_{x} v)(s,\eta_{s}) \right]^2  \right] ds  + O(N^{\alpha+1}\varepsilon^3)\right\}.
\]

As before the term linear in $\varepsilon$ vanishes. Recalling that $\varepsilon = N^{1-\alpha}$ and $\gamma = 3 - \alpha$ we have

\[
\frac{\dd \Pp_{u}^{N}}{\dd \widetilde{\Pp}^{N}}(t)  = \exp \left\{ - M_{t} - \frac{1}{4} N^{\gamma} \int\limits_{0}^{t} \frac{1}{N} \sum\limits_{x=1}^{N-1} \left[ (\nabla_{x} v)(s,\eta_{s}) \right]^2  \, ds  + o(N^{\gamma})\right\}.
\]

Expanding $\left[(\nabla_{x} v)(s,\eta_s)\right]^2 = (v_x(s,\eta_s) - v_{x+1}(s,\eta_s))^2$ leads us to

\begin{align}\label{eq:expansion-of-radon-nikodym}
& \frac{\dd \Pp_{u}^{N}}{\dd \widetilde{\Pp}^{N}}(t)  = \exp \Bigg\{ - M_{t} - \frac{1}{2} N^{\gamma} \int\limits_{0}^{t} \frac{1}{N} \sum\limits_{x=1}^{N} v_x(s,\eta_{s})^2 \, ds  + \frac{1}{2} N^{\gamma} \int\limits_{0}^{t} \frac{1}{N} \sum\limits_{x=1}^{N} v_x(s,\eta_{s})v_{x+1}(s,\eta_{s}) \, ds + o(N^{\gamma})\Bigg\}.
\end{align}

The martingale term will be typically $o(N^{\gamma})$. To see this, we use formula (<ref>) – by performing a calculation similar to the one above we get that

\[
N_{t} = M_{t}^2 - \frac{1}{2} N^{\alpha} \int\limits_{0}^{t} \sum\limits_{x=1}^{N} \left[ 1 + \varepsilon (\nabla_{x} v)(s,\eta_{s})\right] \left[\log \left( 1 + \varepsilon (\nabla_{x} v)(s,\eta_{s})\right) \right]^2 \, ds
\]

is a local martingale with respect to $\widetilde{\Pp}^{N}$. By expanding the $\log$ terms up to $\varepsilon^2$ we see that the second term above is bounded by $C N^{\alpha+1} \varepsilon^2 = C N^{\gamma}$ for some $C > 0$. In particular $N_{t}$ is bounded from below, so it is a supermartingale. Thus $\E N_T \leq \E N_0 = 0$ and $\E M_T^2 \leq C N^\gamma$, so Chebyshev's inequality implies that $M_T = o(N^\gamma)$ with high probability.

The second sum in the exponent in (<ref>) will be small by invariance of the uniform distribution of colors in the biased process. More precisely, at fixed time $s$ for each $x$ the correlation term $v_{x}(s,\eta_{s})v_{x+1}(s,\eta_{s})$ has mean $0$, since $\eta_{s}$ has stationary distribution and by Proposition <ref> in stationarity velocities at different sites are independent with mean $0$. Moreover, for the same reason these terms are uncorrelated for different $x$, so by the weak law of large numbers we get that for any $s \leq T$ and $\delta > 0$

\[
\widetilde{\Pp}^{N} \left( \left| \frac{1}{N} \sum\limits_{x=1}^{N} v_{x}(s,\eta_{s})v_{x+1}(s,\eta_{s}) \right| > \delta \right) \to 0
\]

as $N \to \infty$.

Since this holds for any fixed $s$ and the random variables are bounded, we also have

\[
\widetilde{\Pp}^{N} \left( \left| \int\limits_{0}^{T} \frac{1}{N} \sum\limits_{x=1}^{N} v_{x}(s,\eta_{s})v_{x+1}(s,\eta_{s}) \, ds \right| > \delta \right) \to 0
\]

as $N \to \infty$, which proves that the correlation term is $o(N^{\gamma})$ with high probability. Together with the bound on $M_t$ this proves the desired formula for the Radon-Nikodym derivative.

We can now use Lemma <ref> and the law of large numbers established in Theorem <ref> to prove a large deviation lower bound for the interchange process. As the formula from the lemma suggests, the large deviation rate function will be related to the energy of the process to which the biased interchange process converges.

Recall from (<ref>) that for any process $\pi \in \PP$ its energy was defined by

\[
I(\pi) = \E_{\gamma \sim \pi} \EE(\gamma),
\]

where $\EE(\gamma)$ is the Dirichlet energy of the path $\gamma$ defined by (<ref>). We have the following large deviation lower bound

Let $\Pp^{N}$ be the law of the unbiased interchange process $\eta^N$ and let $\mu^{\eta^{N}}$ be the (random) distribution of the corresponding permutation process $X^{\eta^N}$. Let $P = (X, \Phi)$ be the colored trajectory process associated to the equation (<ref>) and let $\mu$ denote the distribution of $X$. For any open set $\Oo \subseteq \PP$ such that $\mu \in \Oo$ we have

\[
\liminf_{N \to \infty} N^{-\gamma} \log \Pp^{N}\left(\mu^{\eta^{N}} \in \Oo\right) \geq - I(\mu).
\]

It will be enough to show the bound above for $\Oo$ being any open ball $B(\mu, \varepsilon)$ in $\PP$ around $\mu$. Let $\Pp_{u}^{N}$ be the distribution of the unbiased process with colors, $\nu^{\eta^{N}}$ the distribution of the colored permutation process $P^{\eta^N} = (X^{\eta^N}, \Phi^{\eta^N})$ associated to $\eta^N$. Let $\nu$ denote the distribution of $P = (X, \Phi)$ and $\widetilde{B}(\nu, \varepsilon)$ an open ball around $\nu$ in $\MM(\DDD)$. Since the projection $(X, \Phi) \mapsto X$ is continuous as a map from $\DDD$ to $\DD$, the corresponding projection from $\MM(\DDD)$ to $\MM(\DD)$ is also continuous. As $\mu^{\eta^N}$ has the same law under $\Pp^N$ and $\Pp_{u}^N$ (remember that in the latter process the colors do not influence the dynamics of swaps), we have that for any $\varepsilon > 0$ there exists $\varepsilon' > 0$ such that
$\Pp^{N}\left(\mu^{\eta^{N}} \in B(\mu, \varepsilon) \right) \geq \Pp_{u}^{N}\left(\nu^{\eta^{N}} \in \widetilde{B}(\nu, \varepsilon')\right)$. Thus to prove the large deviation bound it is sufficient to prove the local lower bound

\begin{equation}\label{eq:lower-bound-nu-epsilon}
\liminf\limits_{\varepsilon \to 0}\liminf_{N \to \infty} N^{-\gamma} \log \Pp_{u}^{N}\left(\nu^{\eta^{N}} \in \widetilde{B}(\nu, \varepsilon)\right) \geq - I(\mu).
\end{equation}

Recall that $\widetilde{\Pp}^{N}$ denotes the distribution of the biased process associated to (<ref>) and consider the Radon-Nikodym derivative $\frac{\dd \Pp_{u}^{N}}{\dd \widetilde{\Pp}^{N}}(t)$.
By Lemma <ref> we have

\begin{equation}\label{eq:radon-nikodym-in-lower-bound}
\frac{\dd \Pp_{u}^{N}}{\dd \widetilde{\Pp}^{N}}(T) = \exp \Bigg\{ -\frac{1}{2} N^{\gamma} \Bigg[ \int\limits_{0}^{T} \frac{1}{N} \sum\limits_{x=1}^{N} v_{x}(\eta_{s})^2 \, ds  +  Y_N\Bigg]\Bigg\},
\end{equation}

where $Y_N$ goes to $0$ in probability as $N \to \infty$.

Now by the law of large numbers from Theorem <ref> and Remark <ref> the distributions $\nu^{\eta^{N}}$ converge in probability in the $d_{\mathcal{W}}^{sup}$ metric to $\nu$ when $\eta^{N}$ is sampled according to $\widetilde{\Pp}^{N}$. Thus for any $\varepsilon > 0$ and an open ball $\widetilde{B}_{\varepsilon} = \{ \zeta \in \MM(\DDD) \, | \, d_{\mathcal{W}}^{sup}(\zeta, \nu) < \varepsilon \}$ around $\nu$ in the $d_{\mathcal{W}}^{sup}$ metric we have $\lim\limits_{N \to \infty} \widetilde{\Pp}^{N}\left(\nu^{\eta^{N}} \in \widetilde{B}_{\varepsilon}\right) = 1$. Since convergence in $d_{\mathcal{W}}^{sup}$ implies convergence in $d_{\mathcal{W}}$, to prove (<ref>) it is enough to analyze the probability $\Pp^N_{u}\left(\nu^{\eta^{N}} \in \widetilde{B}_{\varepsilon}\right)$.

Fix arbitrary $\delta > 0$ and let $U_{N} = \{ |Y_N| \leq \delta \}$. Let $V_{N,\varepsilon} = U_{N} \cap \{ \nu^{\eta^{N}} \in \widetilde{B}_{\varepsilon}\}$ and $\frac{\dd \Pp_{u}^{N}}{\dd \widetilde{\Pp}^{N}} = \frac{\dd \Pp_{u}^{N}}{\dd \widetilde{\Pp}^{N}}(T)$. With $\E$ denoting the expectation with respect to $\Pp_{u}^{N}$ and $\widetilde{\E}$ with respect to $\widetilde{\Pp}^{N}$ we have for any $\varepsilon > 0$ and sufficiently large $N$

\[
\Pp^{N}_{u}\left(\nu^{\eta^{N}} \in \widetilde{B}_{\varepsilon}\right) = \E \left(\id_{\{ \nu^{\eta^{N}} \in \widetilde{B}_{\varepsilon} \}}\right) \geq \E (\id_{V_{N,\varepsilon}}) = \widetilde{\E} \left( \frac{\dd \Pp_{u}^{N}}{\dd \widetilde{\Pp}^{N}} \id_{V_{N,\varepsilon}} \right) \geq \widetilde{\Pp}^{N}(V_{N,\varepsilon}) \left(\inf_{ V_{N,\varepsilon}}\frac{\dd \Pp_{u}^{N}}{\dd \widetilde{\Pp}^{N}} \right).
\]

We have $\lim\limits_{N \to \infty} \widetilde{\Pp}^{N}(V_{N,\varepsilon}) = 1 $ and on the event $U_N$ we have

\[
\frac{\dd \Pp_{u}^{N}}{\dd \widetilde{\Pp}^{N}}(T) \geq \exp \Bigg\{ -\frac{1}{2} N^{\gamma} \Bigg[ \int\limits_{0}^{T} \frac{1}{N} \sum\limits_{x=1}^{N} v_{x}(\eta_{s})^2 \, ds  +  \delta\Bigg]\Bigg\}.
\]

This implies

\begin{equation}\label{eq:lower-bound-with-in}
N^{-\gamma} \log \Pp_{u}^{N}\left(\nu^{\eta^{N}} \in \widetilde{B}_{\varepsilon}\right) \geq - \inf_{\eta \in V_{N,\varepsilon}} I_N (\eta) - \delta,
\end{equation}

\[
I_{N}(\eta) = \frac{1}{2}  \left(\frac{1}{N} \sum\limits_{x=1}^{N} \int\limits_{0}^{T} v_x(s,\eta_{s})^2 \, ds \right).
\]

Now it is not difficult to see that the infimum on the right hand side of (<ref>) converges to $I(\mu)$ as $N \to \infty$ and then $\varepsilon \to 0$. When $(X, \Phi)$ is sampled from $\nu$, $X$ is the solution of (<ref>) with a uniformly random initial condition, so the energy $I(\mu)$ is simply equal to

\[
\E \int\limits_{0}^{T} V(s, X(s),\Phi(s))^2 \, ds,
\]

where the expectation is with respect to the choice of $(X(0), \Phi(0))$. Recall the notation $X_{i}(\eta^{N}_{t}) = \frac{1}{N} x_{i}(\eta^{N}_{t})$, $\Phi_{i}(\eta^{N}_{t}) = \frac{1}{N} \phi_{i}(\eta^{N}_{t})$. In light of (<ref>) what we need to show is that

\begin{equation}\label{eq:energy}
\inf\limits_{\eta \in V_{N,\varepsilon}} \left( \frac{1}{N} \sum\limits_{i=1}^{N} \int\limits_{0}^{T} V(s, X_{i}(\eta^{N}_{s}),\Phi_{i}(\eta^{N}_{s}))^2 \, ds \right) \to \E \int\limits_{0}^{T} V(s, X(s),\Phi(s))^2 \, ds
\end{equation}

as $N \to \infty$ and then $\varepsilon \to 0$.

Consider the trajectory $\eta^N$ and for any particle $i$ let $(X_{i}(t), \Phi_{i}(t))$ denote the solution of (<ref>) corresponding to the initial condition $(X_{i}(\eta^{N}_{0}), \Phi_{i}(\eta^{N}_{0}))$. Since the velocities $V$ are bounded, we can write

\begin{align}
& \left| \int\limits_{0}^{T} \left[ V(s,X_{i}(\eta^{N}_{s}),\Phi_{i}(\eta^{N}_{s}))^2 - V(s,X_{i}(s),\Phi_{i}(s))^2 \right] ds \right| \leq \nonumber \\
& \leq C \int\limits_{0}^{T} \big| V(s,X_{i}(\eta^{N}_{s}),\Phi_{i}(\eta^{N}_{s})) - V(s,X_{i}(s),\Phi_{i}(s)) \big| \, ds \label{eq:speeds-squared} \leq \\ & \leq K T \max \left\{\sup_{t \leq T} \left| X_{i}(\eta^{N}_{t}) - X_{i}(t)\right|, \sup_{t \leq T} \left| \Phi_{i}(\eta^{N}_{t}) - \Phi_{i}(t)\right| \right\} \nonumber
\end{align}

for some $C, K > 0$ depending on the bound on $V$ and the Lipschitz constant of $V$.

Now note that if $\nu^{\eta^N} \in \widetilde{B}_{\varepsilon}$, then by considering the same coupling as in the proof of Theorem <ref> we have

\begin{equation}
\limsup\limits_{N \to \infty} \frac{1}{N} \sum\limits_{i=1}^{N} \left| \max \left\{\sup_{t \leq T} \left| X_{i}(\eta^{N}_{t}) - X_{i}(t)\right|, \sup_{t \leq T} \left| \Phi_{i}(\eta^{N}_{t}) - \Phi_{i}(t)\right| \right\} \right| \leq \varepsilon'
\end{equation}

for some $\varepsilon' > 0$ satisfying $\varepsilon' \to 0$ as $\varepsilon \to 0$. Since $\{ \nu^{\eta^{N}} \in \widetilde{B}_{\varepsilon}\} \subseteq V_{N,\varepsilon}$, combining this with (<ref>) we obtain that the left hand side of (<ref>) converges to

\[ \frac{1}{N} \sum\limits_{i=1}^{N} \int\limits_{0}^{T} V(s,X_{i}(s),\Phi_{i}(s))^2 \, ds
\]

as $N \to \infty$ and then $\varepsilon \to 0$.

Since $(X_{i}(t), \Phi_{i}(t))$ is a solution of (<ref>) and $V$ is the derivative of $X$, the integral is equal simply to the energy of the path $X_{i}(t)$. Since for each $i$ the initial condition $\Phi_{i}(\eta^{N}_{0})$ has uniform distribution on $\left \{\frac{1}{N}, \ldots, 1  \right\}$, independently for all $i$, it follows easily that this expression converges with high probability to the expected energy on the right hand side of (<ref>). This implies $\inf_{\eta \in V_{N,\varepsilon}} I_N (\eta) \to I(\nu)$ as $N \to \infty$ and then $\varepsilon \to 0$. Since in (<ref>)  we can take $\delta$ to be arbitrarily small, this proves (<ref>) and finishes the proof of the lower bound.

With this theorem the large deviation lower bound for generalized solutions to Euler equations, announced as Theorem <ref> in the introduction, is now an easy corollary.

Let $\Pp^{N}$ be the law of the interchange process $\eta^N$ and let $\mu^{\eta^{N}}$ be the (random) distribution of the corresponding permutation process $X^{\eta^N}$. Let $\pi$ be a permuton process which is a generalized solution to Euler equations (<ref>). Provided $\pi$ satisfies Assumptions (<ref>), for any open set $\Oo \subseteq \PP$ such that $\pi \in \Oo$ we have

\[
\liminf_{N \to \infty} N^{-\gamma} \log \Pp^{N}\left(\mu^{\eta^{N}} \in \Oo\right) \geq - I(\pi).
\]

Let $\pi^{\beta, \delta}$ be the distribution of the process $X^{\beta, \delta}$ defined in Section <ref>. By the first part of Proposition <ref> we have $d_{\mathcal{W}}^{sup}(\pi, \pi^{\beta, \delta}) \to 0$ as first $\delta$ and then $\beta \to 0$, in particular for small enough $\delta$ and $\beta$ we have $\pi^{\beta, \delta} \in \Oo$. Then Theorem <ref> implies that

\[
\liminf_{N \to \infty} N^{-\gamma} \log \Pp^{N}\left(\mu^{\eta^{N}} \in \Oo\right) \geq - I(\pi^{\beta, \delta}).
\]

Since by the second part of Proposition <ref> we have $\lim\limits_{\beta \to 0} \lim\limits_{\delta \to 0} I(\pi^{\beta, \delta}) = I(\pi)$, the lower bound is proved.

Large deviation upper bound

In this section we prove Theorem <ref>, a large deviation upper bound for the distribution of the interchange process (we will drop the term “unbiased” from now on). As a first step we will bound the probability that after a (possibly short) time $t > 0$ we see a fixed permutation in the interchange process. This is summarized in the following

Let $\Pp^{N}$ be the law of the interchange process, with $\eta = \eta^N$ denoting the trajectory of the process. Let $\sigma^N \in \mathcal{S}_N$ be a sequence of permutations. For any $t > 0$ we have

\[
\limsup\limits_{N \to \infty} N^{-\gamma}\log \Pp^{N}(\eta_{0}^{-1}\eta_{t} = \sigma_N) \leq - \frac{1}{t} \left(\liminf\limits_{N \to \infty} I(\sigma^N) \right),
\]

where $I(\sigma)$ is the energy of the permutation $\sigma$ defined in (<ref>).

In other words, the large deviation rate of seeing a permutation $\sigma$ at time $t$ in the interchange process is asymptotically bounded from above by $\frac{1}{t}$ times the energy of the permutation $\sigma$.

To prove Proposition <ref> we will employ exponential martingales. The idea is as follows – if $M_{S}(\eta)$ is a function of the process (depending on some set of parameters $S$) which is a positive mean one martingale, then for any permutation $\sigma \in \mathcal{S}_N$ we can write

\begin{align}\label{eq:upper-bound-ej}
& \Pp^{N}(\eta_{0}^{-1}\eta_{t} = \sigma) = \E (\id_{\{\eta_{0}^{-1}\eta_{t} = \sigma\}} )= \E \left( M_{S}(\eta) M_{S}(\eta)^{-1} \id_{\{\eta_{0}^{-1}\eta_{t} = \sigma\}}\right) \leq \\
& \sup_{ \{\chi_{0}^{-1}\chi_{t} = \sigma\}} M_{S}(\chi)^{-1} \E \left( M_{S}(\eta) \id_{\{\eta_{0}^{-1}\eta_{t} = \sigma\}}\right) \leq \sup_{\{\chi_{0}^{-1}\chi_{t} = \sigma\}} M_{S}(\chi)^{-1}, \nonumber
\end{align}

where the supremum is over all deterministic permutation-valued paths $\chi = (\chi_s, 0 \leq s \leq T)$ satisfying $\chi_{0}^{-1}\chi_{t} = \sigma$ and the last inequality comes from the fact that $M_{S}(\chi)$ is a positive mean one martingale. If $M_{S}$ depends only on the increment $\chi_{0}^{-1}\chi_{t}$, we obtain a particularly simple expression

\[
\Pp^{N}(\eta_{0}^{-1}\eta_{t} = \sigma) \leq M_{S}(\sigma)^{-1}.
\]

We can then optimize over the set of parameters $S$ to obtain a large deviation upper bound. The family of martingales we will use is similar to the one used in analyzing large deviations for a simple random walk.

Fix $t > 0$ and a sequence $S = (s_{1}, \ldots, s_{N})$, with $s_{i} \in \left\{ \frac{-1 + \frac{1}{N}}{t}, \frac{-1 + \frac{2}{N}}{t}, \ldots, \frac{1 - \frac{2}{N}}{t}, \frac{1 - \frac{1}{N}}{t} \right\}$. We will think of $s_i$ as “velocity” assigned to the particle $i$. Consider the function

\[
F_{S}(\eta_{t}) = \varepsilon \sum\limits_{i=1}^{N} s_{i} x_{i}(\eta_{t}),
\]

where $x_{i}(\eta_{t})$ is the position of the particle $i$ in the configuration $\eta_{t}$. If $\Ll$ is the generator of the interchange process, given by (<ref>), then by the formula (<ref>) for exponential martingales we obtain that

\[
M_{t}^{S} = \exp \left\{ F_{S}(\eta_{t}) - F_{S}(\eta_{0}) - \int\limits_{0}^{t} e^{-F_{S}(\eta_{s})} \Ll e^{F_{S}(\eta_{s})} \, ds \right\}
\]

is a mean one positive martingale with respect to $\Pp^{N}$.

For simplicity we will use the same notation $s_{x}(\eta) = s_{\eta^{-1}(x)}$ as for velocities $v_x$ of particles in the previous sections (with the convention that $i$ denotes labels of particles and $x$ denotes the positions), although bear in mind that now $s_x$ are just parameters, not related in any way to the the biased interchange process considered in the preceding sections. We have

\[
\Ll e^{F_{S}(\eta)} = \frac{1}{2} N^{\alpha} \sum\limits_{x=1}^{N-1} \left( e^{F_{S}(\eta^{x,x+1})} - e^{F_{S}(\eta)}  \right) = \frac{1}{2} N^{\alpha} \sum\limits_{x=1}^{N-1} \left( e^{F_{S}(\eta) + \varepsilon \left[s_{x}(\eta) - s_{x+1}(\eta)\right]} - e^{F_{S}(\eta)}  \right),
\]

\[
M_{t}^{S} = \exp \left\{ \varepsilon \sum\limits_{i=1}^{N} s_{i} \left( x_{i}(\eta_{t}) - x_{i}(\eta_{0})\right) - \frac{1}{2} N^{\alpha} \int\limits_{0}^{t} \sum\limits_{x=1}^{N-1} \left( e^{\varepsilon [s_{x}(\eta_{s}) - s_{x+1}(\eta_{s})]} -1\right) ds \right\}.
\]

Expanding up to order $\varepsilon^2$ we get

\begin{align}\label{eq:exponential}
M_{t}^{S} = & \exp \Bigg\{ \varepsilon \sum\limits_{i=1}^{N} s_{i} \left( x_{i}(\eta_{t}) - x_{i}(\eta_{0})\right) - \frac{1}{2} N^{\alpha} \varepsilon \int\limits_{0}^{t} \sum\limits_{x=1}^{N-1} [s_{x}(\eta_{s}) - s_{x+1}(\eta_{s})]  \, ds - \\
& -  \frac{1}{4} N^{\alpha}\varepsilon^2 \int\limits_{0}^{t} \sum\limits_{x=1}^{N-1} \left(s_{x}(\eta_{s}) - s_{x+1}(\eta_{s})\right)^2  \, ds  + O(N^{\alpha+1} \varepsilon^3 ) \Bigg\} \nonumber,
\end{align}

where the constants in the $O(\cdot)$ notation depend on $t$ (which is fixed). Observe that the sum of $s_{x} - s_{x+1}$ telescopes, leaving only terms with $s_{1}$ and $s_{N}$, which are $O(N^{\alpha} \varepsilon) = o(N^{\gamma})$. Rescaling by appropriate powers of $N$ and expressing the exponents in terms of the large deviation exponent $\gamma$ we get

\begin{equation*}
M_{t}^{S} = \exp \Bigg\{ N^{\gamma} \left[ \frac{1}{N}\sum\limits_{i=1}^{N} s_{i} \left( \frac{x_{i}(\eta_{t}) - x_{i}(\eta_{0})}{N} \right) -  \frac{1}{4} \int\limits_{0}^{t} \frac{1}{N}\sum\limits_{x=1}^{N-1} \left(s_{x}(\eta_{s}) - s_{x+1}(\eta_{s})\right)^2  \, ds  \right]+ o(N^{\gamma}) \Bigg\}.
\end{equation*}

Expanding $(s_x - s_{x+1})^2$ we obtain (after adding and subtracting the boundary terms $s_1^2$, $s_N^2$ which are only $o(1)$ after rescaling) twice the sum of $s_{x}^{2}$ and the sum of mixed terms $s_x s_{x+1}$. Since $\sum\limits_{x=1}^N s_x^2 = \sum\limits_{i=1}^N s_i^2 $ does not depend on time, we can write

\begin{align*}
M_{t}^{S} = & \exp \Bigg\{ N^{\gamma} \Bigg[ \frac{1}{N}\sum\limits_{i=1}^{N} s_{i} \left( \frac{x_{i}(\eta_{t}) - x_{i}(\eta_{0})}{N} \right) -  \frac{t}{2} \left( \frac{1}{N}\sum\limits_{i=1}^{N} s_{i}^{2} \right) + \\
& + \frac{1}{2} \int\limits_{0}^{t} \frac{1}{N}\sum\limits_{x=1}^{N-1} s_{x}(\eta_{s}) s_{x+1}(\eta_{s}) \, ds + o(1)\Bigg] \Bigg\}.
\end{align*}

As in the proof of the law of large numbers we want to use the one block estimate to get rid of the sum involving correlations between $s_{x}$ for adjacent $x$. This time the correlation term might not be small, since $s_{i}$ are arbitrary, but typically it will be nonnegative, so we can neglect it for the sake of the upper bound. More precisely, we have the following

Let $\Pp^{N}$ be the law of the interchange process. Fix $t > 0$ and let
$s_{i} \in \left\{ \frac{-1}{t}, \frac{-1 + \frac{1}{N}}{t}, \ldots, \frac{1 - \frac{1}{N}}{t}, \frac{1}{t} \right\}$. Then, with notation as above, we have for any $\delta > 0$

\[
\limsup\limits_{N \to \infty} N^{-\gamma} \log \Pp^{N} \left( \int\limits_{0}^{t} \frac{1}{N}\sum\limits_{x=1}^{N-1} s_{x}(\eta_{s}) s_{x+1}(\eta_{s}) \, ds \leq - \delta \right) = - \infty.
\]

We employ Lemma <ref> with $w(x,\phi) = \frac{2x-1}{t}$, $a_i = s_i$ and $b_{x}(\eta) = a_{x}(\eta)$, in particular $h_{x}(\eta) = s_x(\eta) s_{x-1}(\eta)$. As in the lemma consider $\E_{\mu_{x,l}^{\eta_s}} \left(s_{x}(\eta) s_{x+1}(\eta)\right)$, where $\mu_{x,l}^{\eta_s}$ is the empirical distribution of $a_i$ in a box $\Lambda_{x,l}$. Let us write

\begin{align}\label{eq:sxsx+1}
& \int\limits_{0}^{t} \frac{1}{N}\sum\limits_{x=1}^{N-1} s_{x}(\eta_s) s_{x+1}(\eta_s) \, ds = \\
& = \int\limits_{0}^{t} \frac{1}{N}\sum\limits_{x=1}^{N-1} \left( s_{x}(\eta_s) s_{x+1}(\eta_s) - \E_{\mu_{x,l}^{\eta_s}} \left[s_{x}(\eta) s_{x+1}(\eta)\right] \right)\, ds + \int\limits_{0}^{t} \frac{1}{N}\sum\limits_{x=1}^{N-1} \E_{\mu_{x,l}^{\eta_s}} \left[s_{x}(\eta) s_{x+1}(\eta) \right]\, ds. \nonumber
\end{align}

Since under $\mu_{x,l}^{\eta_s}$ the colors are i.i.d. random variables, we have $\E_{\mu_{x,l}^{\eta_s}} \left[s_{x}(\eta) s_{x+1}(\eta)\right] = \left( \E_{\mu_{x,l}^{\eta_{s}}}  s_{x}(\eta)\right )^2 \geq 0$, so the second term on the right hand side of (<ref>) is nonnegative for every $l$. Lemma <ref> guarantees that for any $\delta > 0$

\[
\limsup\limits_{l \to \infty} \limsup\limits_{N \to \infty} N^{-\gamma} \log \Pp^{N} \left( \int\limits_{0}^{t} \frac{1}{N}\sum\limits_{x=1}^{N-1} \left( s_{x}(\eta_s) s_{x+1}(\eta_s) - \E_{\mu_{x,l}^{\eta_s}} \left[s_{x}(\eta) s_{x+1}(\eta)\right] \right)\, ds  \leq - \delta \right) = - \infty.
\]

Since the left hand side of (<ref>) does not depend on $l$, this finishes the proof.

With this lemma the proof of Proposition <ref> is rather straightforward.

Lemma <ref> implies that for any $a > 0$ there exist sets $\Oo_{N,a}$ such that on $\Oo_{N,a}$ we have

\begin{equation}\label{eq:upper-bound-mt}
M_{t}^{S}(\eta) \geq \exp \Bigg\{ N^{\gamma} \Bigg[ \frac{1}{N}\sum\limits_{i=1}^{N} s_{i} \left( \frac{x_{i}(\eta_{t}) - x_{i}(\eta_{0})}{N} \right) -  \frac{t}{2} \left( \frac{1}{N}\sum\limits_{i=1}^{N} s_{i}^{2} \right) - a + o(1)\Bigg] \Bigg\},
\end{equation}

with the $o(1)$ term depending on $t$, and $\Pp^{N}(\Oo_{N,a}^{c}) \to 0 $ as $N \to \infty$ superexponentially fast

\[
\limsup_{N \to \infty} N^{-\gamma} \log \Pp^{N}(\Oo_{N,a}^{c}) = - \infty.
\]

Now we can use the strategy outlined earlier with the positive mean one martingale $M_{t}^{S}(\eta)$. We write

\begin{align*}
& \Pp^{N}(\eta_{0}^{-1}\eta_{t} = \sigma^N) = \E \left(\id_{\{\eta_{0}^{-1}\eta_{t} = \sigma^N\}} \right)= \E \left( M_{t}^{S}(\eta)^{-1} M_{t}^{S}(\eta)  \id_{\{\eta_{0}^{-1}\eta_{t} = \sigma^N\}}\right) = \\
& \E \left( M_{t}^{S}(\eta)^{-1} M_{t}^{S}(\eta)  \id_{\{\eta_{0}^{-1}\eta_{t} = \sigma^N \}}\id_{\Oo_{N,a}} \right) + \E \left(  \id_{\{\eta_{0}^{-1}\eta_{t} = \sigma^N \}}\id_{\Oo_{N,a}^{c}} \right).
\end{align*}

On $\Oo_{N,a}$ we can use the bound (<ref>) obtained above. Note also that on the event $\{\eta_{0}^{-1}\eta_{t} = \sigma^N\}$ we have $x_{i}(\eta_{t}) - x_{i}(\eta_{0}) = \sigma^N(i) - i$, which together with (<ref>) leads us to

\begin{equation}\label{eq:one-slice-intermediate}
\Pp^{N}(\eta_{0}^{-1}\eta_{t} = \sigma^N) \leq e^{-N^{\gamma} \left(I_{S}(\sigma^N) - a + o(1)\right)} + \Pp^{N}(\Oo_{N,a}^{c}),
\end{equation}

\[
I_{S}(\sigma^N) = \frac{1}{N}\sum\limits_{i=1}^{N} s_{i} \left( \frac{\sigma^N(i) - i}{N} \right) -  \frac{t}{2} \left( \frac{1}{N}\sum\limits_{i=1}^{N} s_{i}^{2} \right).
\]

To optimize over the choice of $S = (s_1, \ldots, s_N)$, observe that $I_{S}(\sigma^N)$ is quadratic in $s_{i}$, so an easy calculation shows that the optimal choice is

\[
s_{i} = \frac{\sigma^N(i) - i}{t N},
\]

which is valid, since we assumed $s_{i} \in \left\{ \frac{-1 + \frac{1}{N}}{t}, \frac{-1 + \frac{2}{N}}{t}, \ldots, \frac{1 - \frac{1}{N}}{t}\right\}$. This gives the maximal value of $I_{S}(\sigma^N)$ equal to

\[
\frac{1}{2} \left( \frac{1}{N}\sum\limits_{i=1}^{N} \frac{1}{t}\left( \frac{\sigma^N(i) - i}{N} \right)^2 \right),
\]

which is exactly the energy $I(\sigma^N)$ rescaled by $t$. Inserting this into (<ref>) gives us

\[
\Pp^{N}(\eta_{0}^{-1}\eta_{t} = \sigma^N) \leq e^{-N^{\gamma} \left(I(\sigma^N) - a + o(1)\right)} + \Pp^{N}(\Oo_{N,a}^{c})
\]

Since $\limsup\limits_{n \to \infty} \frac{1}{n} \log(a_n + b_n) = \max\{\limsup\limits_{n \to \infty} \frac{1}{n} \log a_n, \limsup\limits_{n \to \infty} \frac{1}{n} \log b_n\}$, we obtain

\[
\limsup\limits_{N \to \infty} N^{-\gamma} \log \Pp^{N}(\eta_{0}^{-1}\eta_{t} = \sigma^N) \leq \max\left\{ - \liminf\limits_{N \to \infty} I(\sigma^N) + a, \limsup\limits_{N \to \infty} N^{-\gamma} \log \Pp^{N} (\Oo_{N,a}^{c}) \right\}.
\]

The second lim sup is $-\infty$ and by taking $a \to 0$ we arrive at

\[
\limsup\limits_{N \to \infty} N^{-\gamma} \log \Pp^{N}(\eta_{0}^{-1}\eta_{t} = \sigma^N) \leq - \liminf\limits_{N \to \infty} I(\sigma^N)
\]

as desired.

We can readily extend the bound from Proposition <ref> to all finite-dimensional distributions of the interchange process as follows. Fix a finite set of times $0 \leq t_0 < t_1 < \ldots < t_k \leq T$ and for clarity of notation let us write $\vec{\eta}_{t_0, \ldots, t_k} = (\eta_{t_{0}}^{-1}\eta_{t_1}, \ldots, \eta_{t_{k-1}}^{-1}\eta_{t_k})$ for the corresponding sequence of increments of $\eta$. Suppose we want to bound the probability $\Pp^{N}(\vec{\eta}_{t_0, \ldots, t_k} = (\sigma^N_{1}, \ldots, \sigma^N_{k}))$, where $(\sigma^N_{1}, \ldots, \sigma^N_{k})$ is a fixed sequence of permutations for each $N$, $\sigma^N_j \in \mathcal{S}_N$. Recall that the interchange process has independent increments, i.e., the permutations $\left(\eta_{t_{j}}\right)^{-1} \eta_{t_{j+1}}$ for any family non-overlapping intervals $[t_{j}, t_{j+1})$ are independent. Therefore we can write

\[
\Pp^{N}(\vec{\eta}_{t_0, \ldots, t_k} = (\sigma^N_{1}, \ldots, \sigma^N_{k})) = \prod\limits_{j=1}^{k} \Pp^{N}(\eta_{t_{j-1}}^{-1} \eta_{t_j} = \sigma^N_{j}).
\]

As the interchange process is stationary, we have $\Pp^{N}(\eta_{t_{j-1}}^{-1} \eta_{t_j} = \sigma^N_{j}) = \Pp^{N}(\eta_{0}^{-1} \eta_{t_{j} - t_{j-1}} = \sigma^N_{j})$. Thus by applying Proposition <ref> we obtain

\begin{equation}\label{eq:many-slices}
\limsup\limits_{N \to \infty} N^{-\gamma}\log \Pp^{N}(\vec{\eta}_{t_0, \ldots, t_k} = (\sigma^N_{1}, \ldots, \sigma^N_{k})) \leq - \liminf\limits_{N \to \infty} \sum\limits_{j=1}^{k} \frac{1}{t_j - t_{j-1}}I(\sigma^N_j).
\end{equation}

Recall that $\mu^{\eta^N}$ denotes the distribution of the random permutation process associated to $\eta^N$ (defined by (<ref>)) and for a finite partition $\Pi$ by $I^{\Pi}(\mu^{\eta^N})$ we denote the approximation of energy of $\mu^{\eta^N}$ associated to $\Pi$ (defined by (<ref>)). From equation (<ref>) we obtain the following corollary which will be useful later

For any $C > 0$ and any finite partition $\Pi = \{ 0 = t_0 < t_1 < \ldots < t_k = T\}$ we have

\[
\limsup\limits_{N \to \infty} N^{-\gamma}\log \Pp^{N}(I^{\Pi}(\mu^{\eta^{N}}) \geq C) \leq - C.
\]

Consider the set $A_{C}^{N}$ of all sequences of permutations $(\sigma^N_1, \ldots, \sigma^N_k)$, $\sigma^N_j \in \mathcal{S}_N$, such that $\sum\limits_{j=1}^{k} \frac{1}{t_j - t_{j-1}}I(\sigma^N_j) \geq C$. By performing a union bound over all such sequences we get

\[
\Pp^{N}(I^{\Pi}(\mu^{\eta^{N}}) \geq C) \leq N!^k \sup\limits_{A_{C}^{N}} \Pp^{N}(\vec{\eta}_{t_0, \ldots, t_k} = (\sigma^N_{1}, \ldots. \sigma^N_{k})),
\]

Now it is enough to observe that for fixed $k$ we have $\log (N!^k) = o(N^\gamma)$ and apply (<ref>).

Now we can proceed to prove a general large deviation upper bound, announced as Theorem <ref> in the introduction,. Recall that $\PP \subseteq \MM(\DD)$ denotes the space of all permuton and approximate permuton processes.

Let $\Pp^{N}$ be the law of the interchange process $\eta^{N}$ and let $\mu^{\eta^{N}}$ be the (random) distribution of the corresponding random permutation process $X^{\eta^N}$. For any closed set $\mathcal{C} \subseteq \PP$ we have

\[
\limsup_{N \to \infty} N^{-\gamma} \log \Pp^{N}\left(\mu^{\eta^{N}} \in \mathcal{C} \right) \leq - \inf\limits_{\pi \in \mathcal{C}} I(\pi),
\]

where $I(\pi)$ is the energy of the process $\pi$ defined by (<ref>).

It is standard (see, e.g., <cit.>) that the large deviation upper bound for closed sets follows from a local upper bound for open balls and exponential tightness of the sequence $\mu^{\eta^{N}}$. The exponential tightness part will be proved in Proposition <ref> below, so here we focus on the first part, that is, we will prove that for any $\pi \in \PP$ we have

\begin{equation}\label{eq:ld-ball}
\limsup_{\varepsilon \to 0} \limsup_{N \to \infty} N^{-\gamma} \log \Pp^{N}\left(\mu^{\eta^{N}} \in B(\pi, \varepsilon) \right) \leq - I(\pi),
\end{equation}

where $B(\pi, \varepsilon)$ denotes the open $\varepsilon$-ball around $\pi$ in the Wasserstein distance $d_{\mathcal{W}}$ on $\PP$.

Fix a finite set of times $0 = t_{0} < t_{1} < \ldots < t_{k} = T$. Since almost surely the interchange process does not make jumps at any of the prescribed times $t_0, t_1, \ldots, t_k$, by continuity of projections for any $\varepsilon > 0$ there exists $\varepsilon' > 0$ such that

\begin{equation}\label{eq:upper-bound-projection}
\Pp^{N}\left( d_{\mathcal{W}}(\mu^{\eta^{N}}, \pi) < \varepsilon' \right) \leq \Pp^{N} \left(d(\mu^{\eta^{N}}_{t_{0}, t_{1}}, \pi_{t_0, t_1}) < \varepsilon) \wedge \ldots \wedge d(\mu^{\eta^{N}}_{t_{k-1}, t_{k}},\pi_{t_{k-1}, t_{k}}) < \varepsilon \right),
\end{equation}

where $d$ denotes the Wasserstein distance on $\MM([0,1]^2)$. Furthermore, note that the permutation process with distribution $\mu^{\eta^{N}}$ has independent increments, i.e., the permutations $\left(\eta_{t_{j}}^{N}\right)^{-1} \eta_{t_{j+1}}^{N}$ for any family non-overlapping intervals $[t_{j}, t_{j+1})$ are independent. Thus we can write

\begin{align}\label{eq:upper-bound-product}
\Pp^{N} \left(d(\mu^{\eta^{N}}_{t_{0}, t_{1}}, \pi_{t_0, t_1}) < \varepsilon) \wedge \ldots \wedge d(\mu^{\eta^{N}}_{t_{k-1}, t_{k}},\pi_{t_{k-1}, t_{k}}) < \varepsilon \right) = \prod\limits_{i=0}^{k-1} \Pp^{N} \left( d(\mu^{\eta^{N}}_{t_{i}, t_{i+1}}, \pi_{t_{i}, t_{i+1}}) < \varepsilon)\right).
\end{align}

In this way we have reduced the problem to bounding the probability that the random measure $\mu^{\eta^{N}}_{t_{i}, t_{i+1}}$ is close to a fixed permuton $\pi_{t_{i}, t_{i+1}}$.

Fix $i$ and consider all permutations $\sigma \in \mathcal{S}_{N}$ such that the empirical measure $\mu_{\sigma}$ satisfies $d(\mu_{\sigma}, \pi_{t_{i}, t_{i+1}}) < \varepsilon$. As there are at most $N!$ such permutations, by performing a union bound over this set we obtain

\[
\Pp^{N} \left( d(\mu^{\eta^{N}}_{t_{i}, t_{i+1}}, \pi_{t_{i}, t_{i+1}}) < \varepsilon)\right) \leq N! \sup_{ \substack{\sigma \in \mathcal{S}_{N} \\ d(\mu_{\sigma}, \pi_{t_{i}, t_{i+1}}) < \varepsilon}} \Pp^{N} \left(\mu^{\eta^{N}}_{t_{i}, t_{i+1}} = \mu_{\sigma}\right),
\]

where on the right hand side we have the probability that the random measure $\mu_{t_i, t_{i+1}}^{\eta^{N}}$ is equal to $\mu_{\sigma}$. This probability is simply equal to $\Pp^{N} \left( \left(\eta_{t_i}^{N}\right)^{-1} \eta_{t_{i+1}}^{N} = \sigma)\right)$ and by stationarity of the interchange process this is the same as $\Pp^{N} \left( \left(\eta_{0}^{N}\right)^{-1} \eta_{t_{i+1} - t_i}^{N} = \sigma)\right)$.
By employing Proposition <ref>, with $\sigma^N \in \mathcal{S}_N$ being any permutation attaining the supremum above, and noticing that $\log N! = o(N^{\gamma})$ we get

\[
\limsup_{N \to \infty} N^{-\gamma} \log \Pp^{N} \left( d(\mu^{\eta^{N}}_{t_i, t_{i+1}}, \pi_{t_{i}, t_{i+1}}) < \varepsilon)\right) \leq \limsup_{N \to \infty} \left(- \frac{1}{t_{i+1} - t_i} I(\sigma^N)\right).
\]

Now observe that for any $\sigma$ such that $d(\mu_{\sigma}, \pi_{t_{i}, t_{i+1}}) < \varepsilon$ the energy $I(\sigma) = I(\mu_{\sigma})$ has to be close to $I(\pi_{t_{i}, t_{i+1}})$, the energy of the permuton $\pi_{t_i, t_{i+1}}$ (recall definition <ref>), since $I$ is continuous in the weak topology on $\MM([0,1]^2)$. Thus upon taking $\varepsilon \to 0$ we obtain

\[
\limsup_{\varepsilon \to 0} \limsup_{N \to \infty} N^{-\gamma} \log \Pp^{N} \left(d(\mu^{\eta^{N}}_{t_i, t_{i+1}}, \pi_{t_{i}, t_{i+1}}) < \varepsilon \right) \leq -\frac{1}{t_{i+1} - t_{i}} I(\pi_{t_{i}, t_{i+1}}) .
\]

Applying this estimate to the product in (<ref>) and observing that in (<ref>) without loss of generality we can assume $\varepsilon' \leq \varepsilon$, we arrive at the following bound

\begin{align*}
& \limsup_{\varepsilon \to 0} \limsup_{N \to \infty} N^{-\gamma} \log \Pp^{N}\left(d_{\mathcal{W}}(\mu^{\eta^{N}}, \pi) < \varepsilon \right) \leq -\sum\limits_{i=0}^{k-1} \frac{1}{t_{i+1} - t_{i}} I(\pi_{t_{i}, t_{i+1}}).
\end{align*}

Since $t_0, t_1, \ldots, t_k$ were arbitrary, by optimizing over all finite partitions $\Pi = \{ 0 = t_{0} < t_{1} < \ldots < t_{k} = T \}$ we obtain

\[
\limsup_{\varepsilon \to 0} \limsup_{N \to \infty} N^{-\gamma} \log \Pp^{N}\left(d_{\mathcal{W}}(\mu^{\eta^{N}}, \pi) < \varepsilon \right) \leq - \sup_{\Pi} \sum\limits_{i=0}^{k-1} \frac{1}{t_{i+1} - t_{i}} I(\pi_{t_{i}, t_{i+1}}).
\]

Recalling the definitions (<ref>), (<ref>) and (<ref>), to prove (<ref>) it remains to show that we have $I(\pi) = \sup\limits_{\Pi} I^{\Pi}(\pi)$, which is exactly the statement of Lemma <ref>.

The family of measures $\mu^{\eta^{N}}$ is exponentially tight, that is, there exists a sequence of compact sets $K_{m} \subseteq \PP$ such that

\[
\limsup_{N \to \infty} N^{-\gamma} \log \Pp^{N} (\mu^{\eta^{N}} \notin K_{m}) \leq - m.
\]

The idea of the proof is to show that having many particles whose trajectories have poor modulus of continuity (which would spoil compactness) necessarily implies the process having high energy, which by Corollary <ref> is unlikely.

Recall that $\DD = \DD([0,T], [0,1])$ is the space of all càdlàg paths from $[0,T]$ to $[0,1]$. It will be convenient to work with the following notion of càdlàg modulus of continuity – for a path $f \in \DD$ we define

\[
w_{\delta}''(f) = \sup_{\substack{t_1 \leq t \leq t_2 \\ t_2 - t_1 \leq \delta}} \left\{ |f(t) - f(t_1)| \wedge |f(t_2) - f(t)| \right\}.
\]

By a characterization of compactness in the Skorokhod space (<cit.>) a set $A \subseteq \DD$ has compact closure if and only if the following conditions hold

\[
\begin{cases}
\sup\limits_{f \in A} \sup\limits_{t \in [0,T]} |f(t)| < \infty, \\
\lim\limits_{\delta \to 0} \sup\limits_{f \in A} w_{\delta}''(f) = 0, \\
\lim\limits_{\delta \to 0} \sup\limits_{f \in A} |f(\delta) - f(0)| = 0, \\
\lim\limits_{\delta \to 0} \sup\limits_{f \in A} |f(T-) - f(T-\delta)| = 0.
\end{cases}
\]

In our setting the first condition is trivially satisifed. To exploit the other conditions let us introduce for any $m,r \geq 1$ the following sets

\begin{align*}
& K^{w}_{m,r} = \bigcap\limits_{k \geq 1}\left\{ f \in \DD \, \big\vert \, w_{\delta_{k}(m,r)}''(f) \leq \varepsilon_{k} \right\}, \\
& K^{0}_{m,r} = \bigcap\limits_{k \geq 1}\left\{ f \in \DD \, \big\vert \, |f(\delta_{k}(m,r)) - f(0)| \leq \varepsilon_{k} \right\}, \\
& K^{T}_{m,r} = \bigcap\limits_{k \geq 1}\left\{ f \in \DD \, \big\vert \, |f(T-) - f(T - \delta_{k}(m,r))| \leq \varepsilon_{k} \right\},
\end{align*}

\[
K_{m,r} = K^{w}_{m,r} \cap K^{0}_{m,r} \cap K^{T}_{m,r},
\]

where $\varepsilon_{k} = 4^{-k}$ and $\delta_{k}(m,r)$ will be appropriately chosen later. We will assume that for fixed $m,r$ we have $\lim\limits_{k \to \infty} \delta_{k}(m,r) = 0$ and that for any $k \geq 1$ both $\frac{T}{\delta_k}$ and $\frac{\delta_{k}(m,r)}{\delta_{k+1}(m,r)}$ are integer (the latter assumption is for simplicity of notation only). Note that by the aforementioned compactness conditions each set $K_{m,r}$ has compact closure in $\DD$.

\[
K_m = \bigcap\limits_{r \geq 1} \left\{ \mu \in \MM(\DD) \, \bigg\vert \, \mu(K_{m,r}) \geq 1 - \frac{1}{r} \right\}.
\]

We claim that $K_m$ has compact closure in $\MM(\DD)$. Indeed, by Prokhorov's theorem it is enough to prove that $K_m$ is tight. If $\mu \in K_m$, then for any $r \geq 1$ we have $\mu(K_{m,r}^{c}) \leq \frac{1}{r}$, so the sets $K_{m,r}$ form the family of compact sets needed for tightness of $K_m$.

The sets $K_m$ (possibly after taking their closures) will form the family of compact sets needed for exponential tightness. Thus our goal is to bound $\Pp^{N}(\mu^{\eta^{N}} \notin K_{m})$. Let us write

\[
\Pp^{N}(\mu^{\eta^{N}} \notin K_{m}) = \Pp^{N}\left(\exists_{r \geq 1} \, \, \mu^{\eta^{N}} (K_{m,r}^{c}) \geq \frac{1}{r} \right) \leq \sum\limits_{r \geq 1} \Pp^{N}\left(\mu^{\eta^{N}} (K_{m,r}^{c}) \geq \frac{1}{r}\right).
\]

It is enough to show that for any $m,r \geq 1$ and any $N \geq 1$ we have

\begin{equation}\label{eq:exp-tightness-m}
\Pp^{N}\left(\mu^{\eta^{N}} (K_{m,r}^{c}) \geq \frac{1}{r} \right) \leq C e^{-mr N^{\gamma}},
\end{equation}

where $C > 0$ is some global constant.

For any given $m$ and $r$, observe that $\mu^{\eta^{N}} (K_{m,r}^{c}) \geq \frac{1}{r}$ means that in $\eta^N$ we have at least $\frac{N}{r}$ particles with paths $f \notin K_{m,r}$. Since $K_{m,r} = K^{w}_{m,r} \cap K^{0}_{m,r} \cap K^{T}_{m,r}$, clearly it is enough to estimate separately the probabilities that at least $\frac{N}{3r}$ particles have paths respectively not in $K^{w}_{m,r}$, $K^{0}_{m,r}$ or $K^{T}_{m,r}$. The argument for $K^{0}_{m,r}$ and $K^{T}_{m,r}$ is much simpler, so we skip it and concentrate only on the case of $K^{w}_{m,r}$. For simplicity we will write $\alpha(r) = \frac{1}{3r}$

For fixed $m$ and $r$ we will call a path $f$ bad if $w_{\delta_{k}(m,r)}''(f) > \varepsilon_{k}(m,r)$ for some $k \geq 1$. We will call $f$ bad exactly at scale $k$ if $w_{\delta_{k}(m,r)}''(f) > \varepsilon_{k}(m,r)$, but $w_{\delta_{j}(m,r)}''(f) \leq \varepsilon_{j}(m,r)$ for all $j \geq k+1$. Recalling the definition of the set $K^{w}_{m,r}$, the event whose probability we would like to bound is

\[
A_{N}^{m,r} = \left\{ \mbox{there exist} \geq \alpha(r) N \mbox{ particles with bad paths} \right\}.
\]

Consider now the events

\[
B_{N}^{m,r,k} = \left\{ \mbox{there exist} \geq \frac{\alpha(r)}{2^k} N \mbox{ particles whose paths are bad exactly at scale $k$} \right\}.
\]

Note that if $f$ is a bad path with jumps of fixed size $\frac{1}{N}$, then there exists $k \geq 1$ such that $f$ is bad exactly at scale $k$ (since all paths we are considering are cádlàg). Thus we have $A_{N}^{m,r} \subseteq \bigcup\limits_{k \geq 1} B_{N}^{m,r,k}$, so

\[
\Pp^{N}\left(\mu^{\eta^{N}} ((K_{m,r}^{w})^c) \geq \alpha(r)\right) = \Pp^{N}\left(A_{N}^{m,r}\right) \leq \sum\limits_{k \geq 1} \Pp^{N}(B_{N}^{m,r,k}).
\]

Thus it is enough to show that for any $m,r,k \geq 1$ and any $N \geq 1$ we have

\begin{equation}\label{eq:bound-on-b}
\Pp^{N}(B_{N}^{m,r,k}) \leq e^{-mrk N^\gamma}.
\end{equation}

From now on we fix $m,r,k$ and $N$. All paths we are considering are assumed to come from the interchange process $\eta^N$, in particular they have jumps of fixed size $\frac{1}{N}$. For the sake of brevity we will simply write $\delta_k = \delta_k(m,r)$.

Let us divide the interval $[0,T]$ into $J = \frac{T}{\delta_k}$ intervals of the form $[j \delta_k, (j+1)\delta_k]$, $j=0, \ldots, J-1$. Consider any path $f$ which is bad exactly at scale $k$. The condition $w_{\delta_{k}}''(f) > \varepsilon_{k}$ implies that for some $t, t_1, t_2$ such that $t ,t_2 \in [t_1, t_1 + \delta_k]$ we have $|f(t) - f(t_1)| > \varepsilon_k$ and $|f(t_2) - f(t)| > \varepsilon_k$. A simple application of the triangle inequality implies that there exists $j \in \{0, \ldots, J-1 \}$ and $t' \in [j \delta_k, (j+1)\delta_k]$ such that $|f(j\delta_k) - f(t')| > \frac{\varepsilon_k}{2}$.

Let us consider the interval $[s, s'] = [j \delta_k, (j+1)\delta_k]$ obtained above. Subdivide it into $L = \frac{\delta_k}{\delta_{k+1}}$ intervals of the form $[s_{\ell}, s_{\ell+1}]$, $\ell = 0, \ldots, L-1$, where $s_{\ell} = s + \ell \delta_{k+1}$. For $\ell=0, \ldots, L-1$ let $\Delta_{\ell}(f) = |f(s_{\ell}) - f(s_{\ell+1})|$.

The crucial observation is that $\sum\limits_{\ell=0}^{L-1} \Delta_{\ell}(f) > \frac{\varepsilon_k}{4}$. To see this, consider $t'$ such that $|f(s) - f(t')| > \frac{\varepsilon_k}{2}$, obtained above, and let $\tilde{\ell}$ be such that $t' \in (s_{\tilde{\ell}}, s_{\tilde{\ell} + 1}]$. By the triangle inequality we have

\[
\frac{\varepsilon_{k}}{2} < |f(s) - f(t')| \leq \sum\limits_{\ell=0}^{\tilde{\ell} - 2} \Delta_{\ell}(f) + |f(s_{\tilde{\ell}}) - f(t')|.
\]

Since $f$ is bad exactly at scale $k$, we have $w_{\delta_{k+1}}''(f) \leq \varepsilon_{k+1}$, which together with $|s_{\tilde{\ell}} - t'| \leq \delta_{k+1}$ implies $|f(s_{\tilde{\ell}}) - f(t')| \leq \varepsilon_{k+1} = 4^{-(k+1)} = \frac{\varepsilon_{k}}{4}$. Thus necessarily $\sum\limits_{\ell=0}^{\tilde{\ell} - 2} \Delta_{\ell}(f) > \frac{\varepsilon_k}{4}$. From this we obtain

\begin{equation}\label{eq:sum-of-delta}
\frac{\varepsilon_k^2}{16} < \left( \sum\limits_{\ell=0}^{L-1} \Delta_{\ell}(f) \right)^2 \leq L \sum\limits_{\ell=0}^{L-1} \Delta_{\ell}(f)^2,
\end{equation}

where the right-hand side estimate follows from the Cauchy-Schwarz inequality.

Now let us suppose that the event $B_{N}^{m,r,k}$ holds. Then there exist at least $\frac{\alpha(r)}{2^k}N$ paths $f_i$ for which the estimate (<ref>) holds. Consider the partition $\Pi = \{0 = t_0 < t_1 < \ldots < t_n = T \}$ where $n = \frac{T}{\delta_{k+1}}$, $t_j = j \delta_{k+1}$ for $j=0, \ldots, n$. Recalling that $f_i = \frac{1}{N}\eta^{N}(i)$, the definition of $\Delta_{\ell}(f)$ and the definition (<ref>) of the energy $I^{\Pi}(\mu^{\eta^{N}})$ we obtain that on $B_{N}^{m,r,k}$ we have

\begin{align*}
I^{\Pi}(\mu^{\eta^{N}}) =  &\frac{1}{N} \sum\limits_{i=1}^{N} \left( \frac{1}{2} \sum\limits_{j=1}^{n} \frac{| f_{i}(t_{j}) - f_{i}(t_{j-1}) |^2}{t_{j} - t_{j-1}} \right) = \\
& \frac{1}{N} \sum\limits_{i=1}^{N} \left(\frac{1}{2\delta_{k+1}} \sum\limits_{j=1}^{n} | f_{i}(t_{j}) - f_{i}(t_{j-1}) |^2 \right) >
\frac{1}{N} \cdot \frac{\alpha(r)}{2^k}N \cdot \left( \frac{1}{2\delta_{k+1}} \frac{\varepsilon_k^2}{16 L} \right) = \\
& \frac{\alpha(r)}{2^{k+5}} \frac{1}{\delta_{k+1}} \frac{\varepsilon_k^2}{ \frac{\delta_k}{\delta_{k+1}}} = \frac{\varepsilon_k^2}{\delta_k} \frac{\alpha(r)}{2^{k+5}}.
\end{align*}

Writing again $\delta_k = \delta_k(m,r)$, we have thus obtained the bound

\[
\Pp^{N}(B_{N}^{m,r,k}) \leq \Pp^{N}\left( I^{\Pi}(\mu^{\eta^{N}}) \geq \frac{\varepsilon_k^2}{\delta_k(m,r)} \frac{\alpha(r)}{2^{k+5}}\right).
\]

Recalling $\varepsilon_k = 4^{-k}$, $\alpha(r) = \frac{1}{3r}$, we see that to prove (<ref>) it is sufficient to take $\delta_k(m,r)$ small enough so that

\[
\frac{4^{-2k}}{\delta_k(m,r)} \frac{1}{3r 2^{k+5}} \geq 2 mrk.
\]

By applying Corollary <ref> we obtain that

\[
\Pp^{N}(B_{N}^{m,r,k}) \leq \Pp^{N}\left( I^{\Pi}(\mu^{\eta^{N}}) \geq 2mrk \right)\leq e^{-mrkN^{\gamma}}
\]

for $N$ large enough. By taking $\delta_k(m,r)$ even smaller if necessary we can make this estimate true for all values of $N \geq 1$, which proves (<ref>) and finishes the proof of exponential tightness.

Asymptotics of relaxed sorting networks

In this section we prove the limiting behavior of random relaxed sorting networks, given by Theorem <ref>, and the asymptotic counting formula of Theorem <ref>. With the large deviation bounds obtained in the preceding sections both of the proofs are now rather straightforward.

Let $\mathcal{R} \subseteq \PP$ be the set of permuton processes $X$ reaching exactly the reverse permuton at time $1$, i.e., such that $(X_0, X_1) \sim (X, 1 - X)$, and likewise let $\mathcal{R}_N$ be the set of permutation processes on $N$ elements reaching exactly the reverse permutation $\mathrm{rev}_N = (N \, \ldots \, 2 \, 1)$ at time $1$. Let $\mathcal{R}_{\delta}$ denote the $\delta$-neighborhood in the Wasserstein distance on $\PP$ of the set $\mathcal{R} \cup \bigcup\limits_{N \geq 1} \mathcal{R}_N$.

Let $\eta^N$ be the interchange process with $\alpha = 1 + \kappa \in (1,2)$ and let $\mu^{\eta^N}$ be the distribution of the corresponding permutation process. By definition of a random relaxed sorting network, for any given $\delta > 0$ we have for sufficiently large $N$

\[
\Pp^N \left( \pi^{N}_{\delta} \in B(\pi_{\mathcal{A}}, \varepsilon) \right) = \Pp^N \left( \mu^{\eta^N} \in B(\pi_{\mathcal{A}}, \varepsilon) \big\vert \mu^{\eta^N} \in \mathcal{R}_{\delta} \right).
\]

Now, we have

\[
\Pp^N \left( \mu^{\eta^N} \notin B(\pi_{\mathcal{A}}, \varepsilon) \big\vert \mu^{\eta^N} \in \mathcal{R}_{\delta} \right) = \frac{1}{\Pp^N\left( \mu^{\eta^N} \in \mathcal{R}_{\delta} \right)}\Pp^N \left( \left\{ \mu^{\eta^N} \notin B(\pi_{\mathcal{A}}, \varepsilon) \right\} \cap \left\{ \mu^{\eta^N} \in \mathcal{R}_{\delta} \right\} \right).
\]

By the large deviation lower bound of Theorem <ref> we have

\[
\Pp^N\left( \mu^{\eta^N} \in \mathcal{R}_{\delta} \right) \geq \exp \left\{-N^\gamma \left( I(\pi_{\mathcal{A}}) + o(1) \right) \right\},
\]

where $\gamma = 3 - \alpha$.

Let $\mathcal{C}_{\varepsilon, \delta} = B(\pi_{\mathcal{A}}, \varepsilon)^{c} \cap \overline{\mathcal{R}_{\delta}}$. By the large deviation upper bound of Theorem <ref> we have

\[
\Pp^N \left( \mu^{\eta^N} \in \mathcal{C}_{\varepsilon, \delta} \right) \leq \exp \left\{ -N^\gamma \left( \inf\limits_{\mu \in \mathcal{C}_{\varepsilon, \delta}} I(\mu) + o(1)\right) \right\}.
\]

Since $\pi_{\mathcal{A}}$ is the unique minimizer of energy on $\mathcal{R}$ ([6], [18]), given $\varepsilon > 0$ there exists $\beta = \beta(\varepsilon) > 0$ such that

\begin{equation}\label{eq:infimum-sine}
\inf\limits_{\mu \in B(\pi_{\mathcal{A}}, \varepsilon)^c \cap \mathcal{R}} I(\mu) \geq I(\pi_{\mathcal{A}}) + \beta.
\end{equation}

Since $I(\cdot)$ is lower semi-continuous, by (<ref>) we obtain (possibly after adjusting $\delta$ to replace $\overline{\mathcal{R}_{\delta}}$ with $\mathcal{R}_{\delta}$) that for all sufficiently small $\delta$ we have

\[
\inf\limits_{\mu \in B(\pi_{\mathcal{A}}, \varepsilon)^c \cap \mathcal{R}_{\delta}} I(\mu) \geq I(\pi_{\mathcal{A}}) + \frac{\beta}{2}.
\]

Altogether we obtain that for all sufficiently small $\delta$ (depending on $\varepsilon$ only)

\[
\Pp^N \left( \mu^{\eta^N} \notin B(\pi_{\mathcal{A}}, \varepsilon) \big\vert \mu^{\eta^N} \in \mathcal{R}_{\delta} \right) \leq e^{N^{\gamma} \left( I(\pi_{\mathcal{A}}) + o(1) \right)} e^{- N^{\gamma} \left( I(\pi_{\mathcal{A}}) + \frac{\beta}{2} + o(1) \right)} = e^{-N^{\gamma} \left( \frac{\beta}{2} + o(1) \right)}
\]

and the right-hand side goes to $0$ as $N \to \infty$

Let $\Pp^N$ denote the law of the interchange process with $\alpha = 1 + \kappa$. Let $J$ be the number of all particle swaps in the process and let $M = \left\lfloor \frac{1}{2} N^{\alpha}(N-1) \right\rfloor$.

Let $\mathcal{S}_{\delta}$ denote the $\delta$-neighborhood of the reverse permuton in the Wasserstein distance on $\MM([0,1]^2)$. Observe that for sufficiently large $N$ we have for any $k \leq M$

\begin{equation}\label{eq:comparison-with-k}
\Pp^N \left( \mu^{\eta^N}_{0,T} \in \mathcal{S}_{\delta} \big\vert J = k\right) \leq \Pp^N \left( \mu^{\eta^N}_{0,T} \in \mathcal{S}_{\delta} \big\vert J = M\right)(1 + o(1)).
\end{equation}

This is because if the process has done $k$ swaps up to time $T_k$ and $\mu^{\eta^N}_{0,T_k} \in \mathcal{S}_{\delta}$, then with high probability $\mu^{\eta^N}_{0,T_M} \in \mathcal{S}_{\delta}$ as well, since $\mathcal{S}_{\delta}$ is an open set in $\MM([0,1]^2)$ and the additional number of steps done between $T_k$ and $T_M$ is $\leq \frac{1}{2} N^{\alpha}(N-1) = o(N^3)$, so typically almost all particles have negligible displacement.

On the other hand, since in the interchange process each sequence of swaps of given length is equally likely, we have

\[
\Pp^N \left( \mu^{\eta^N}_{0,T} \in \mathcal{S}_{\delta} \big\vert J = M\right) = \frac{|\mathcal{S}^{N}_{\kappa, \delta}|}{|\PP^{N}_{M}|},
\]

where $\PP^{N}_{M}$ is the set of all sequences of adjacent transpositions of length $M$. Summing (<ref>) over $k \leq M$ we obtain

\[
\Pp^N \left( \mu^{\eta^N}_{0,T} \in \mathcal{S}_{\delta} \cap \{ J \leq M \}\right) \leq \Pp^N \left( J \leq M \right) \frac{|\mathcal{S}^{N}_{\kappa, \delta}|}{|\PP^{N}_{M}|}(1 + o(1)).
\]

Since under $\Pp^N$ $J$ has Poisson distribution with mean $\frac{1}{2}N^{\alpha}(N-1)$, we have $\Pp^N \left( J \leq M \right) \to 1/2$ as $N \to \infty$.

To estimate the left-hand side, let $\widetilde{\Pp}^N$ be the law of the biased interchange process corresponding to the sine curve process $\pi_{\mathcal{A}}$. Recall Lemma <ref> and for fixed $\varepsilon > 0$ let $A$ be the event that the $o(1)$ term in the formula for $\frac{\dd \Pp_{u}^{N}}{\dd \widetilde{\Pp}^{N}}(T)$ is at most $\varepsilon$.
Let us write

\[
\Pp^N \left( \mu^{\eta^N}_{0,T} \in \mathcal{S}_{\delta} \cap \{ J \leq M \} \cap A\right) \leq \Pp^N \left( \mu^{\eta^N}_{0,T} \in \mathcal{S}_{\delta} \cap \{ J \leq M \}\right)
\]

\begin{align*}
& \Pp^N \left( \mu^{\eta^N}_{0,T} \in \mathcal{S}_{\delta} \cap \{ J \leq M \} \cap A \right) = \\
& = \widetilde{\Pp}^N \left( \mu^{\eta^N}_{0,T} \in \mathcal{S}_{\delta} \cap \{ J \leq M \} \cap A \right) \frac{\Pp^N \left( \mu^{\eta^N}_{0,T} \in \mathcal{S}_{\delta} \cap \{ J \leq M \}\cap A \right)}{\widetilde{\Pp}^N \left( \mu^{\eta^N}_{0,T} \in \mathcal{S}_{\delta} \cap \{ J \leq M \} \cap A \right)}.
\end{align*}

By Theorem <ref> $\mu^{\eta^N}_{0,T} \in \mathcal{S}_{\delta}$ has high probability under $\widetilde{\Pp}^N$ and, since the particle swap rates for the biased process sum up to $\frac{1}{2}N^{\alpha}(N-1)$ (recall (<ref>)), we have similarly as for the unbiased process $\widetilde{\Pp}^N \left( J \leq M \right) \to 1/2$ as $N \to \infty$. By Lemma <ref> $A$ is a high probability event under $\widetilde{\Pp}^N$ as well.

To estimate the remaining probabilities, we employ the formula for the Radon-Nikodym derivative from Lemma <ref>. Since in the biased process with high probability the energy term in the derivative is close to $I(\pi_{\mathcal{A}}) = \frac{\pi^2}{6}$, we obtain

\[
\frac{\Pp^N \left( \mu^{\eta^N}_{0,T} \in \mathcal{S}_{\delta} \cap \{ J \leq M \}\cap A \right)}{\widetilde{\Pp}^N \left( \mu^{\eta^N}_{0,T} \in \mathcal{S}_{\delta} \cap \{ J \leq M \} \cap A \right)} \geq e^{-N^{\gamma} \left( \frac{\pi^2}{6} + \varepsilon \right) + o(N^\gamma)},
\]

where $\gamma = 3 - \alpha$.

Altogether we obtain

\[
|\mathcal{S}^{N}_{\kappa, \delta}| \geq |\PP^{N}_{M}| e^{-N^{\gamma} \left( \frac{\pi^2}{6} + \varepsilon \right) + o(N^\gamma)}.
\]

Since $|\PP^{N}_{M}| = (N-1)^M = e^{\lfloor \frac{1}{2} N^{\alpha}(N-1) \rfloor \log(N-1)}$ and $\varepsilon$ was arbitrary, we obtain the asymptotic lower bound on $|\mathcal{S}^{N}_{\kappa, \delta}|$ as claimed.

For the upper bound, let $\mathcal{R}_{\delta}$ be as in the previous theorem. By the large deviation upper bound of Theorem <ref> we have

\[
\Pp^N\left( \mu^{\eta^N}_{0,T} \in \mathcal{R}_{\delta} \right) \leq \exp \left\{ -N^{\gamma} \left( \inf\limits_{\mu \in \overline{\mathcal{R}_{\delta}}} I(\mu) + o(1) \right) \right\},
\]

Since $I$ is lower semi-continuous, given any $\varepsilon > 0$ we have for all sufficiently small $\delta > 0$

\[
\inf\limits_{\mu \in \overline{\mathcal{R}_{\delta}}} I(\mu) \geq \inf\limits_{\mu \in \mathcal{R}} I(\mu) - \varepsilon = I(\pi_{\mathcal{A}}) - \varepsilon,
\]

where again we have used the energy minimization property of $\pi_{\mathcal{A}}$. Since $I(\pi_{\mathcal{A}}) = \frac{\pi^2}{6}$, this implies that for any $\varepsilon > 0$ and sufficiently small $\delta > 0$

\[
\Pp^N\left( \mu^{\eta^N} \in \mathcal{R}_{\delta} \right) \leq e^{-N^{\gamma} \left( I(\pi_{\mathcal{A}}) - \varepsilon + o(1) \right)}.
\]

Now we estimate

\[
\Pp^N\left( \mu^{\eta^N} \in \mathcal{R}_{\delta} \right) \geq \Pp^N\left( \mu^{\eta^N} \in \mathcal{R}_{\delta} \big\vert J = M\right) \Pp^N \left( J = M \right) = \frac{|\mathcal{S}^{N}_{\kappa, \delta}|}{|\PP^{N}_{M}|}\Pp^N \left( J = M \right)
\]

and use the same asymptotic estimate for $|\PP^{N}_{M}|$ as in the lower bound. Since $J$ is Poisson with mean $\frac{1}{2}N^\alpha (N-1)$ under $\Pp^N$, the second term on the right-hand side is $e^{O(\log N)}$. Altogether we obtain

\[
|\mathcal{S}^{N}_{\kappa, \delta}| \leq \exp\left\{ \frac{1}{2}N^{1 + \kappa} (N-1) \log(N-1)- N^{2 - \kappa} \left( I(\pi_{\mathcal{A}}) - \varepsilon \right) + o(N^{2 - \kappa})\right\},
\]

which proves the desired asymptotic upper bound on $|\mathcal{S}^{N}_{\kappa, \delta}|$.

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Michał Kotowski

Institute of Mathematics of the Polish Academy of Sciences

ul. Śniadeckich 8 00-656 Warszawa, Poland

E-mail<EMAIL_ADDRESS>
Bálint Virág

Department of Mathematics, University of Toronto

40 St George St.
Toronto, ON, M5S 2E4, Canada

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