We turn to the second term in (7.6). By the strong Markov property, it is
bounded by

$\max_{z\in{\mathscr{A}}_{r,k}}{\mathbf{P}}^{n}_{\\!z}\big{[}\,H_{{\mathscr{V}}^{(p)}_{j}}<H_{\breve{{\mathscr{V}}}^{(p)}_{j}}\,\big{]}\;.$

To complete the proof, it remains to show that for all
$z\in{\mathscr{A}}_{r,k}$,

$\lim_{n\to\infty}\frac{\pi_{n}(x)}{\pi_{n}({\mathscr{V}}^{(p)}_{j})}\;{\mathbf{P}}^{n}_{\\!z}\big{[}\,H_{{\mathscr{V}}^{(p)}_{j}}<H_{\breve{{\mathscr{V}}}^{(p)}_{j}}\,\big{]}^{2}\,\;=\;0\;.$

Since $\pi_{n}(x)\prec\pi_{n}(z)$, it is enough to show that

$\lim_{n\to\infty}\frac{\pi_{n}(z)}{\pi_{n}({\mathscr{V}}^{(p)}_{j})}\;{\mathbf{P}}^{n}_{\\!z}\big{[}\,H_{{\mathscr{V}}^{(p)}_{j}}<H_{\breve{{\mathscr{V}}}^{(p)}_{j}}\,\big{]}^{2}\,\;=\;0\;.$
(7.7)

This follows from the induction hypothesis. Indeed, as
$z\in{\mathscr{V}}^{(r+1)}$, either $z$ belongs to ${\mathscr{V}}^{(p)}$ or
$z$ belongs to some ${\mathscr{V}}^{(s)}_{\ell}$, $r<s<p$, for some
$\ell\in{\mathfrak{T}}_{s}$. In the first case, the probability vanishes
because $z\in\breve{{\mathscr{V}}}^{(p)}_{j}$ (as
$z\in{\mathscr{V}}^{(p)}\cap{\mathscr{A}}_{r,k}$ and
${\mathscr{A}}_{r,k}\cap{\mathscr{V}}^{(p)}_{j}=\varnothing$,
$z\in\breve{{\mathscr{V}}}^{(p)}_{j}$). In the second case, (7.7) holds by the
induction hypothesis.

It remains to consider the case where
$\pi_{n}({\mathscr{V}}^{(p)}_{j})\prec\pi_{n}(x)$ and $x\in\Delta$. We repeat
the induction argument. Write (7.6) with ${\mathscr{A}}_{x}$ instead of
${\mathscr{A}}_{r,k}$. We estimate the first term on the right-hand side of
(7.6) as before, applying Lemma 7.7 instead of Lemma 7.6. The second term is
also bounded as before. At the end of the argument one needs to estimate (7.7)
for $z\in{\mathscr{V}}$, $\pi_{n}({\mathscr{V}}^{(p)}_{j})\prec\pi_{n}(z)$.
This has been done in the first part of the proof. ∎

##  8\. Proof of Theorem 3.3

We assume in this section that the dynamics is reversible:
$\pi_{n}(x)\,R_{n}(x,y)=\pi_{n}(y)\,R_{n}(y,x)$ for all $(x,y)\in E$.

### Elementary properties of $\pi_{n}$

The proof of Theorem 3.3 requires some preparation. We first introduce the
transient equivalent classes of the chain $X^{(n)}_{t}$. We say that $y$ is
equivalent to $x$,
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}y\sim x$ if
$y=x$ or if there exists a sequence $x=x_{0},\dots,x_{\ell}=y$,
$y_{0}=y,\dots,y_{m}=x$ such that ${\mathbb{R}}_{0}(x_{i},x_{i+1})>0$,
${\mathbb{R}}_{0}(y_{j},y_{j+1})>0$ for all $0\leq i<\ell$, $0\leq j<m$.

This relation divides the set $V$ into equivalent classes. Clearly the sets
${\mathscr{V}}_{j}$ are equivalent classes, but there might be others. Denote
by
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathscr{C}}_{1},\dots,{\mathscr{C}}_{{\mathfrak{m}}}$
the equivalent classes which _have more than one element_ and are not one of
the sets ${\mathscr{V}}_{j}$, $1\leq j\leq{\mathfrak{n}}$. Note that the sets
${\mathscr{C}}_{1},\dots,{\mathscr{C}}_{{\mathfrak{m}}}$,
${\mathscr{V}}_{1},\dots,{\mathscr{V}}_{{\mathfrak{n}}}$ may not exhaust $V$:
the set $V$ may contain elements which do not belong to one of the
${\mathscr{C}}_{k}$’s nor to one of the ${\mathscr{V}}_{j}$’s.

The first assertion extends (2.16) to the sets ${\mathscr{C}}_{k}$. We claim
that if $x$, $y$ belong to the same class ${\mathscr{C}}_{k}$, then

$\lim_{n\to\infty}\frac{\pi_{n}(x)}{\pi_{n}(y)}\;=\;a\in(0,\infty)\;.$ (8.1)

Indeed, by definition, there exists a sequence $x=x_{0},\dots,x_{\ell}=y$,
such that ${\mathbb{R}}_{0}(x_{i},x_{i+1})>0$, for all $0\leq i<\ell$. By
reversibility,

$\frac{\pi_{n}(x)}{\pi_{n}(y)}\;=\;\frac{R_{n}(x_{\ell},x_{\ell-1})\dots
R_{n}(x_{1},x_{0})}{R_{n}(x_{0},x_{1})\dots
R_{n}(x_{\ell-1},x_{\ell})}\;\cdot$

By hypothesis, the denominator converges to a positive real number. On the
other hand, by (2.1), the numerator converges to a non-negative real number.
This proves that $\pi_{n}(x)/\pi_{n}(y)\to a\in[0,\infty)$. Inverting the
roles of $x$ and $y$ we conclude that $a\in(0,\infty)$, as claimed in (8.1).

Fix an oriented edge $(x,y)\in E$ whose endpoints belong to the same
equivalent class ${\mathscr{V}}^{(1)}_{j}$ or ${\mathscr{C}}_{k}$, $j\in
S_{1}$, $1\leq k\leq{\mathfrak{m}}$. We claim that

$\text{${\mathbb{R}}_{0}(x,y)>0$ if and only if ${\mathbb{R}}_{0}(y,x)>0$}\;.$
(8.2)

Indeed, assume that ${\mathbb{R}}_{0}(x,y)>0$. Since
$\pi_{n}(x)\,R_{n}(x,y)=\pi_{n}(y)\,R_{n}(y,x)$, by (2.16) and (8.1),
$\lim_{n\to\infty}R_{n}(y,x)={\mathbb{R}}_{0}(x,y)\lim_{n\to\infty}\pi_{n}(x)/\pi_{n}(y)>0$.

Denote by
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathbb{L}}^{(0)}_{j}$,
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathbb{L}}^{(0)}_{T,k}$,
$j\in S_{1}$, $1\leq k\leq{\mathfrak{m}}$, the generators associated to the
rates ${\mathbb{R}}_{0}$ restricted to the equivalent classes
${\mathscr{V}}^{(1)}_{j}$, ${\mathscr{C}}_{k}$, respectively. This means that
we set to $0$ all jumps from ${\mathscr{C}}_{k}$ to its complement. Denote by
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\nu_{k}$ the
stationary state of the Markov chain associated to the generator
${\mathbb{L}}^{(0)}_{T,k}$.

We claim that for all $1\leq k\leq{\mathfrak{m}}$,

$\lim_{n\to\infty}\frac{\pi_{n}(x)}{\pi_{n}({\mathscr{C}}_{k})}\;=\;\nu_{k}(x)\quad\text{for
all $x\in{\mathscr{C}}_{k}$ and that $\nu_{k}$ is reversible}\;.$ (8.3)

This result extends Lemma 6.1 to the transient sets ${\mathscr{C}}_{k}$. To
establish (8.3), let $m\in{\mathscr{P}}({\mathscr{C}}_{k})$ be the limit of
the sequence of measures $\pi_{n}(\,\cdot\,)/\pi_{n}({\mathscr{C}}_{k})$. This
limit exists by (8.1). By reversibility, for all $x$, $y\in{\mathscr{C}}_{k}$,

$\frac{\pi_{n}(x)}{\pi_{n}({\mathscr{C}}_{k})}\,R_{n}(x,y)\;=\;\frac{\pi_{n}(y)}{\pi_{n}({\mathscr{C}}_{k})}\,R_{n}(y,x)\;.$

Passing to the limit yields that $m$ satisfies the detailed balance conditions
with respect to ${\mathbb{R}}_{0}$. Hence $m$ is stationary (actually,
reversible), and, by uniqueness, $m=\nu_{k}$. This proves that the sequence of
measures $\pi_{n}(\,\cdot\,)/\pi_{n}({\mathscr{C}}_{k})$ converges to
$\nu_{k}$ and that $\nu_{k}$ is reversible.

The same statement yields that $\pi^{(1)}_{j}$ is a reversible measure for the
chain ${\mathbb{X}}_{t}$ restricted to ${\mathscr{V}}_{j}$, $j\in S_{1}$.

### The functionals ${\mathscr{I}}^{(p)}$

The first result of this section provides an alternative formula for the
functional ${\mathscr{I}}^{(0)}$ introduced in (3.11). Its proof relies on the
construction of a directed graph without directed loops. The equivalence
classes of the chain ${\mathbb{X}}_{t}$ form the set of vertices of this
directed graph. Denote them by ${\mathscr{Q}}_{1},\dots,{\mathscr{Q}}_{\ell}$.
The sets ${\mathscr{V}}_{j}$ and ${\mathscr{C}}_{k}$ belongs to this set and
are vertices of the graph. In other words, for each $j\in S_{1}$, there exists
$1\leq a\leq\ell$ such that ${\mathscr{V}}_{j}={\mathscr{Q}}_{a}$. A similar
statement holds for the sets ${\mathscr{C}}_{k}$.

Draw a directed arrow from ${\mathscr{Q}}_{a}$ to ${\mathscr{Q}}_{b}$ if there
exists $x\in{\mathscr{Q}}_{a}$ and $y\in{\mathscr{Q}}_{b}$ such that
${\mathbb{R}}_{0}(x,y)>0$. Denote the set of directed edges by ${\mathbb{A}}$
and the graph by ${\mathbb{G}}=({\mathbf{Q}},{\mathbb{A}})$, where
${\mathbf{Q}}$ is the set $\\{{\mathscr{Q}}_{1},\dots,{\mathscr{Q}}_{\ell}\\}$
of vertices.

A path in the graph ${\mathbb{G}}=({\mathbf{Q}},{\mathbb{A}})$ is a sequence
vertices $({\mathscr{Q}}_{a_{j}}:0\leq j\leq m)$, such that there is a
directed arrow from ${\mathscr{Q}}_{a_{j}}$ to ${\mathscr{Q}}_{a_{j+1}}$ for
$0\leq j<m$.

This directed graph has no directed loops because the existence of a directed
loop would contradict the definition of the sets ${\mathscr{Q}}_{a}$ as
equivalent classes. (Mind that undirected loops might exist). On the other
hand, since the sets ${\mathscr{V}}_{j}$ are closed irreducible classes, these
sets are not the tail of a directed edge in the graph. Finally, fix an
equivalent class ${\mathscr{Q}}_{a}$ which is not a set ${\mathscr{V}}_{j}$.
Hence, the elements of ${\mathscr{Q}}_{a}$ are transient for the chain
${\mathbb{X}}_{t}$. In particular, there is a path
$({\mathscr{Q}}_{a}={\mathscr{Q}}_{a_{0}},\dots,{\mathscr{Q}}_{a_{m}})$ such
that ${\mathscr{Q}}_{a_{j}}$ is not a closed irreducible class for $0\leq
j<m$, and ${\mathscr{Q}}_{a_{m}}$ is one.

Fix an equivalent class ${\mathscr{Q}}_{a}$ which is not a set
${\mathscr{V}}_{j}$. Denote by ${\mathbf{D}}({\mathscr{Q}}_{a})$ the length of
the longest path from ${\mathscr{Q}}_{a}$ to a closed irreducible class. The
function ${\mathbf{D}}$ is well defined because (a) the set of vertices is
finite, (b) there is at least a path, (c) there are no directed loops in the
graph.

Fix $a$, $b$ such that there is a directed arrow from ${\mathscr{Q}}_{a}$ to
${\mathscr{Q}}_{b}$. Then,

${\mathbf{D}}({\mathscr{Q}}_{a})\;\geq\;{\mathbf{D}}({\mathscr{Q}}_{b})\;+\;1\;.$
(8.4)

Indeed, it is enough to consider the longest path from ${\mathscr{Q}}_{b}$ to
the irreducible classes. ${\mathscr{Q}}_{a}$ does not belong to the path
because there are no directed loops. By adding ${\mathscr{Q}}_{a}$ at the
beginning of the path from ${\mathscr{Q}}_{b}$ to the irreducible classes, we
obtain a path from ${\mathscr{Q}}_{a}$ to the irreducible classes of length
${\mathbf{D}}({\mathscr{Q}}_{b})+1$, proving (8.4).

We may lift the function ${\mathbf{D}}$ to $V$ by setting
${\mathbf{D}}(x)={\mathbf{D}}({\mathscr{Q}}_{a})$ for all
$x\in{\mathscr{Q}}_{a}$.

Let ${\mathscr{J}}^{(0)}\colon{\mathscr{P}}(V)\to[0,+\infty]$ be the
functional defined by

$\displaystyle{\mathscr{J}}^{(0)}(\mu)$ $\displaystyle=\,\sum_{j\in
S_{1}}\big{\langle}\sqrt{f_{j}}\,,\,(-\,{\mathbb{L}}^{(0)}_{j})\sqrt{f_{j}}\,\big{\rangle}_{\pi^{(1)}_{j}}\;+\;\sum_{k=1}^{{\mathfrak{m}}}\big{\langle}\sqrt{g_{k}}\,,\,(-\,{\mathbb{L}}^{(0)}_{T,k})\sqrt{g_{k}}\,\big{\rangle}_{\nu_{k}}$
(8.5)
$\displaystyle+\;\sum_{k=1}^{{\mathfrak{m}}}\sum_{x\in{\mathscr{C}}_{k}}\sum_{y\not\in{\mathscr{C}}_{k}}\mu(x)\,{\mathbb{R}}_{0}(x,y)+\;\sum_{x\not\in{\mathscr{C}}\cup{\mathscr{V}}}\sum_{y\in
V}\mu(x)\,{\mathbb{R}}_{0}(x,y)\;.$

In this formula,
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathscr{C}}=\cup_{k}{\mathscr{C}}_{k}$,
$f_{j}(x)=\mu(x)/\pi^{(1)}_{j}(x)$, $g_{k}(z)=\mu(z)/\nu_{k}(z)$,
$x\in{\mathscr{V}}^{(1)}_{j}$, $z\in{\mathscr{C}}_{k}$.

######  Lemma 8.1.

For every $\mu\in{\mathscr{P}}(V)$,
${\mathscr{I}}^{(0)}(\mu)={\mathscr{J}}^{(0)}(\mu)$.

###### Proof.

Fix $\mu\in{\mathscr{P}}(V)$. We first prove that
${\mathscr{J}}^{(0)}(\mu)\leq{\mathscr{I}}^{(0)}(\mu)$. By definition of the
generator ${\mathbb{L}}^{(0)}$,

${\mathscr{I}}^{(0)}(\mu)\;=\;\sup_{u>0}\,-\,\sum_{(x,y)\in{\mathbb{E}}_{0}}\frac{\mu(x)}{u(x)}\,{\mathbb{R}}_{0}(x,y)\,[\,u(y)-u(x)\,]\;,$
(8.6)

where the sum is performed over all directed edges of ${\mathbb{E}}_{0}$.

Fix $\ell\geq 1$, and define $u_{\ell}:V\to(0,\infty)$ by

$u_{\ell}(x)\;=\;\ell^{{\mathbf{D}}(x)}\;\sqrt{\frac{\mu(x)+\varepsilon}{\pi^{(1)}_{j}(x)}}\;,\quad
u_{\ell}(y)\;=\;\ell^{{\mathbf{D}}(y)}\;\sqrt{\frac{\mu(y)+\varepsilon}{\nu_{k}(y)}}\;,\quad
u_{\ell}(z)\,=\,\ell^{{\mathbf{D}}(z)}\;,$

for $x\in{\mathscr{V}}_{j}$, $j\in S_{1}$, $y\in{\mathscr{C}}_{k}$, $1\leq
k\leq{\mathfrak{m}}$, and $z\not\in{\mathscr{V}}\cup{\mathscr{C}}$. Here,
$\varepsilon=1/\ell$ and guarantees that $u$ is positive. By definition of
${\mathscr{I}}^{(0)}$,

${\mathscr{I}}^{(0)}(\mu)\;\geq\;\limsup_{\ell\to\infty}\,-\,\sum_{(x,y)\in{\mathbb{E}}_{0}}\frac{\mu(x)}{u_{\ell}(x)}\,{\mathbb{R}}_{0}(x,y)\,[\,u_{\ell}(y)-u_{\ell}(x)\,]\;.$
(8.7)

We examine the asymptotic behavior of the right-hand of (8.7). Fix
$(x,y)\in{\mathbb{E}}_{0}$, and suppose, first, that $x$,
$y\in{\mathscr{V}}_{j}$ for some $j\in S_{1}$. In this case, the factors
$\ell^{{\mathbf{D}}}$ cancel, and, as $\ell\to\infty$, the corresponding term
in (8.7) converges to

$\pi^{(1)}_{j}(x)\,\sqrt{f_{j}(x)}\,{\mathbb{R}}_{0}(x,y)\,\big{[}\,\sqrt{f_{j}(y)}\,-\,\sqrt{f_{j}(x)}\,\big{]}$

where $f_{j}(x)=\mu(x)/\pi^{(1)}_{j}(x)$. Therefore, the contributions to the
right-hand side of (8.7), of the sum over the edges $(x,y)\in{\mathbb{E}}_{0}$
such that $x$, $y\in{\mathscr{V}}_{j}$ is

$\big{\langle}\sqrt{f_{j}}\,,\,(-\,{\mathbb{L}}^{(0)}_{j})\sqrt{f_{j}}\,\big{\rangle}_{\pi^{(1)}_{j}}\;.$

The same argument yields that the contributions to the right-hand side of
(8.7), of the sum over the edges $(x,y)\in{\mathbb{E}}_{0}$ such that $x$,
$y\in{\mathscr{C}}_{k}$ for some $1\leq k\leq{\mathfrak{m}}$, is

$\big{\langle}\sqrt{g_{k}}\,,\,(-\,{\mathbb{L}}^{(0)}_{T,k})\sqrt{g_{k}}\,\big{\rangle}_{\nu_{k}}\;,$

where $g_{k}(x)=\mu(x)/\nu_{k}(x)$.

Up to this point we considered all edges $(x,y)\in{\mathbb{E}}_{0}$ whose head
and tail belong to the same equivalent class ${\mathscr{V}}_{j}$ or
${\mathscr{C}}_{k}$. Assume now that this is not the case, and consider the
term

$-\,\frac{\mu(x)}{u_{\ell}(x)}\,{\mathbb{R}}_{0}(x,y)\,[\,u_{\ell}(y)-u_{\ell}(x)\,]\;=\;\mu(x)\,{\mathbb{R}}_{0}(x,y)\;-\;\frac{\mu(x)}{u_{\ell}(x)}\,{\mathbb{R}}_{0}(x,y)\,u_{\ell}(y)\;.$

By definition, and since the measures $\pi^{(1)}_{j}$, $\nu_{k}$ are strictly
positive, $u_{\ell}(y)\leq C_{0}\,\ell^{{\mathbf{D}}(y)}$,
$\mu(x)/u_{\ell}(x)\leq C_{0}\,\ell^{-{\mathbf{D}}(x)}$ for some finite
constant $C_{0}$ independent of $x$, $y$ and $\ell$. The absolute value of the
second term is thus bounded above by
$C_{0}\,\ell^{{\mathbf{D}}(y)-{\mathbf{D}}(x)}$. Since there is an edge from
$x$ to $y$, by (8.4), ${\mathbf{D}}(x)\geq{\mathbf{D}}(y)+1$, which proves
that the second term of the previous displayed equation vanishes as
$\ell\to\infty$.

Fix an edge $(x,y)\in{\mathbb{E}}_{0}$ whose head and tail do not belong to
the same equivalent class ${\mathscr{V}}_{j}$ or ${\mathscr{C}}_{k}$. Since
${\mathscr{V}}_{j}$ is a closed irreducible class, $x\not\in{\mathscr{V}}$.
Suppose that $x\in{\mathscr{C}}_{k}$. Hence, $y\not\in{\mathscr{C}}_{k}$
because they do not belong to the same class. These are the terms which
respond for the third sum in (8.5). the terms in which $x\not\in{\mathscr{C}}$
respond for the fourth sum in (8.5), completing the proof that
${\mathscr{I}}^{(0)}(\mu)\geq{\mathscr{J}}^{(0)}(\mu)$.

We turn to the reverse inequality,
${\mathscr{J}}^{(0)}(\mu)\geq{\mathscr{I}}^{(0)}(\mu)$. By (8.6),

${\mathscr{I}}^{(0)}(\mu)\;\leq\;\sum_{j\in
S_{1}}{\mathscr{I}}^{(0)}_{{\mathscr{V}}_{j}}(\mu)\;+\;\sum_{k=1}^{{\mathfrak{m}}}{\mathscr{I}}^{(0)}_{{\mathscr{C}}_{k}}(\mu)\;+\;{\mathscr{I}}^{(0)}_{{\mathscr{R}}}(\mu)\;,$

where ${\mathscr{I}}^{(0)}_{{\mathscr{V}}_{j}}(\mu)$ is given by formula (8.6)
when the sum is performed over the directed edges $(x,y)$ whose head and tail
belong to ${\mathscr{V}}_{j}$. ${\mathscr{I}}^{(0)}_{{\mathscr{C}}_{k}}(\mu)$
is defined similarly, while ${\mathscr{I}}^{(0)}_{{\mathscr{R}}}(\mu)$
contains the remaining edges.

By [55, Theorem 5],

${\mathscr{I}}^{(0)}_{{\mathscr{V}}_{j}}(\mu)\;=\;\big{\langle}\sqrt{f_{j}}\,,\,(-\,{\mathbb{L}}^{(0)}_{j})\sqrt{f}\,\big{\rangle}_{\pi^{(1)}_{j}}\;,$

where $f_{j}(x)=\mu(x)/\pi^{(1)}_{j}(x)$, $x\in{\mathscr{V}}_{j}$. An
analogous result holds for ${\mathscr{I}}^{(0)}_{{\mathscr{C}}_{k}}(\mu)$.
These two terms correspond to the first two terms in (8.5).

We turn to ${\mathscr{I}}^{(0)}_{{\mathscr{R}}}(\mu)$, which can be written as

${\mathscr{I}}_{{\mathscr{R}}}^{(0)}(\mu)\;=\;\sum_{(x,y)}\mu(x)\,{\mathbb{R}}_{0}(x,y)\;+\;\sup_{u>0}\,-\,\sum_{(x,y)}\frac{\mu(x)}{u(x)}\,{\mathbb{R}}_{0}(x,y)\,u(y)\;.$

where the sums are performed over directed edges whose head and tail belong to
different equivalent classes. Since the second term is negative,

${\mathscr{I}}_{{\mathscr{R}}}^{(0)}(\mu)\;\leq\;\sum_{(x,y)}\mu(x)\,{\mathbb{R}}_{0}(x,y)\;.$

We have seen in the first part of the proof that this sum can be written as
the third and fourth terms in ${\mathscr{J}}^{(0)}(\mu)$, completing the proof
of the lemma. ∎

Note that for each $1\leq k\leq{\mathfrak{m}}$, there exists at least on
$x\in{\mathcal{C}}_{k}$ such that ${\mathbb{R}}_{0}(x,y)>0$ for some
$y\not\in{\mathscr{C}}_{k}$. On the other hand, as the Markov chain associated
to ${\mathbb{L}}^{(0)}_{T,k}$ is ergodic,
$\big{\langle}\sqrt{g}\,,\,(-\,{\mathbb{L}}^{(0)}_{T,k})\sqrt{g}\,\big{\rangle}_{\nu_{k}}=0$
entails that $g$ is constant. Therefore, ${\mathscr{I}}^{(0)}(\mu)\,=\,0$ if
and only if there exists a probability measure $\omega$ on $S_{1}$ such that

$\mu\;=\;\sum_{j\in S_{1}}\omega_{j}\,\pi^{(1)}_{j}\;.$ (8.8)

Fix $1\leq p\leq{\mathfrak{q}}$, and let
${\mathscr{J}}^{(p)}\colon{\mathscr{P}}(V)\to[0,+\infty]$ be the functional
defined as follows. If $\mu=\sum_{j\in S_{p}}\omega_{j}\,\pi^{(p)}_{j}$ for
some probability measure $\omega$ in $S_{p}$, $\omega\in{\mathscr{P}}(S_{p})$,

${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathscr{J}}^{(p)}(\mu)}\,:=\,\sum_{m\in
S_{p+1}}\big{\langle}\sqrt{f_{m}}\,,\,(-\,{\mathbb{L}}^{(p)}_{m})\sqrt{f_{m}}\,\big{\rangle}_{M^{(p)}_{m}}\;+\;\sum_{j\in{\mathfrak{T}}_{p}}\sum_{k\in
S_{p}}\omega_{j}\,r^{(p)}(j,k)\;.$ (8.9)

In this formula,
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathbb{L}}^{(p)}_{m}$
stands for the generator associated to Markov chain ${\mathbb{X}}^{(p)}_{t}$
restricted to the closed irreducible set ${\mathfrak{R}}^{(p)}_{m}$ and
$f_{m}(j)=\omega_{j}/M^{(p)}_{m}(j)$, $j\in{\mathfrak{R}}^{(p)}_{m}$. To
complete the definition of ${\mathscr{J}}^{(p)}$, set

${\mathscr{J}}^{(p)}(\mu)\,:=\,+\infty\quad\text{if $\mu$ is not a convex
combination of $\pi^{(p)}_{j}$, $j\in S_{p}$}\;.$ (8.10)

Recall from (3.13) the definition of the functional
${\mathscr{I}}^{(p)}\colon{\mathscr{P}}(V)\to[0,+\infty]$. The proof of Lemma
8.1 yields that

######  Lemma 8.2.

For all $1\leq p\leq{\mathfrak{q}}$, $\mu\in{\mathscr{P}}(V)$,
${\mathscr{I}}^{(p)}(\mu)={\mathscr{J}}^{(p)}(\mu)$.

Note that, by (8.9) and (3.2), ${\mathscr{I}}^{(p)}(\mu)=0$ if and only if
there exists a probability measure $\widehat{\omega}$ in $S_{p+1}$ such that

$\mu\;=\;\sum_{m\in
S_{p+1}}\widehat{\omega}_{m}\sum_{j\in{\mathfrak{R}}^{(p)}_{m}}M^{(p)}_{m}(j)\,\pi^{(p)}_{j}\;=\;\sum_{m\in
S_{p+1}}\widehat{\omega}_{m}\,\pi^{(p+1)}_{m}\;.$ (8.11)

On the other hand, if $\mu$ is not of this form, by (8.10),
${\mathscr{I}}^{(p+1)}(\mu)$ is set to be equal to $+\infty$. Hence, the
functional ${\mathscr{I}}^{(p+1)}$ is finite only at the $0$-level set of
${\mathscr{I}}^{(p)}$. Furthermore, since the right-hand side of (8.9) is
always finite,

${\mathscr{I}}^{(p+1)}(\mu)\;<\;\infty\quad\text{if and only
if}\quad{\mathscr{I}}^{(p)}(\mu)\;=\;0\;.$ (8.12)

By (8.8), this assertion holds also for $p=0$.

### The $\Gamma$-convergence

We turn to the proof of Theorem 3.3. We proceed by induction. We first show
that ${\mathscr{I}}_{n}$ $\Gamma$-converges to the functional
${\mathscr{I}}^{(0)}$. Then, we observe that, according to (8.8), the
$0$-level set of ${\mathscr{I}}^{(0)}$ corresponds to the convex combinations
of the measures $\pi^{(1)}_{j}$, $j\in S_{1}$. In the sequel, we prove that
$\theta^{(1)}_{n}\,{\mathscr{I}}_{n}$ $\Gamma$-converges to
${\mathscr{I}}^{(1)}$. Clearly, by definition,
${\mathscr{I}}^{(1)}(\mu)=+\infty$ if $\mu$ is not a convex combinations of
the measures $\pi^{(1)}_{j}$, $j\in S_{1}$, while
${\mathscr{I}}^{(1)}(\mu)<+\infty$ if it is. By (8.11), the $0$-level set of
${\mathscr{I}}^{(1)}$ consists of the convex combinations of the measures
$\pi^{(2)}_{j}$, $j\in S_{2}$.

At this point, we iterate the procedure by examining the behavior of
$\theta^{(2)}_{n}\,{\mathscr{I}}_{n}$, and so on until proving that
$\theta^{({\mathfrak{q}})}_{n}\,{\mathscr{I}}_{n}$ $\Gamma$-converges to
${\mathscr{I}}^{({\mathfrak{q}})}$. The $0$-level set of this functional is
the singleton formed by the measure $\pi^{({\mathfrak{q}}+1)}$. As the level
set is a singleton, the iterative procedures ends. Note that this approach
produced the state $\pi^{({\mathfrak{q}}+1)}$ which is is the limit of the
stationary measures $\pi_{n}$:
$\pi^{({\mathfrak{q}}+1)}(x)=\lim_{n\to\infty}\pi_{n}(x)$, $x\in V$.

We turn to the proof that ${\mathscr{I}}_{n}$ $\Gamma$-converges to
${\mathscr{I}}^{(0)}$.

######  Proposition 8.3.

The functional ${\mathscr{I}}_{n}$ $\Gamma$-converges to
${\mathscr{I}}^{(0)}$.

###### Proof.

We start with the $\Gamma-\limsup$. Fix $\mu\in{\mathscr{P}}(V)$ and consider
the sequence $\mu_{n}$ constant equal to $\mu$. By (3.10),

${\mathscr{I}}_{n}(\mu)\;=\;\frac{1}{2}\,\sum_{(x,y)\in
E}\pi_{n}(x)\,R_{n}(x,y)\,\Big{\\{}\sqrt{\frac{\mu(y)}{\pi_{n}(y)}}\,-\,\sqrt{\frac{\mu(x)}{\pi_{n}(x)}}\,\Big{\\}}^{2}\;,$

Fix an edge $(x,y)\in E$. We examine the asymptotic behavior of

$\pi_{n}(x)\,R_{n}(x,y)\,\Big{\\{}\sqrt{\frac{\mu(y)}{\pi_{n}(y)}}\,-\,\sqrt{\frac{\mu(x)}{\pi_{n}(x)}}\,\Big{\\}}^{2}\;.$
(8.13)

By reversibility, this term is symmetric in $x$, $y$.

There are three types of edges. Assume first that $R_{n}(x,y)\to 0$ and
$R_{n}(y,x)\to 0$. By [43, Lemma 3.1], either $\pi_{n}(x)/\pi_{n}(y)$
converges to a nonnegative real number or so does $\pi_{n}(y)/\pi_{n}(x)$.
Assume, without loss of generality because (8.13) is symmetric, that
$\pi_{n}(y)/\pi_{n}(x)\to a\in[0,\infty)$. In this case, (8.13) is equal to

$R_{n}(y,x)\,\Big{\\{}\sqrt{\frac{\mu(x)\,\pi_{n}(y)}{\pi_{n}(x)}}\,-\,\sqrt{\mu(y)}\,\Big{\\}}^{2}\;,$

which vanishes as $n\to\infty$.

Assume that $R_{n}(x,y)\not\to 0$ and $R_{n}(y,x)\to 0$. Hence,
${\mathbb{R}}_{0}(x,y)>0$, ${\mathbb{R}}_{0}(y,x)=0$. In particular, as the
set ${\mathscr{V}}_{j}$ are closed irreducible classes,
$x\not\in{\mathscr{V}}$ (if $x\in{\mathscr{V}}_{j}$ and
${\mathbb{R}}_{0}(x,y)>0$, then $y\in{\mathscr{V}}_{j}$ because it is a closed
irreducible class. Hence, by (8.2), ${\mathbb{R}}_{0}(y,x)>0$, which is a
contradiction). Two possibilities remain, either $x\in{\mathscr{C}}_{k}$ for
some $k$ or $x\not\in{\mathscr{C}}$.

By reversibility, $\pi_{n}(x)/\pi_{n}(y)\to 0$. Hence, (8.13) is equal to

$R_{n}(x,y)\,\Big{\\{}\sqrt{\frac{\mu(y)\,\pi_{n}(x)}{\pi_{n}(y)}}\,-\,\sqrt{\mu(x)}\,\Big{\\}}^{2}\;,$

which converges to $\mu(x)\,{\mathbb{R}}_{0}(x,y)$. If
$x\in{\mathscr{C}}_{k}$, by (8.2), $y\not\in{\mathscr{C}}_{k}$. These are the
pairs which appear in the third term on the right-hand side of (8.5). If
$x\not\in{\mathscr{C}}$ these pairs are responsible for the fourth term on the
right-hand side of (8.5).

Finally, assume that $R_{n}(x,y)\not\to 0$ and $R_{n}(y,x)\not\to 0$. This
means that $x$ and $y$ belong to the same equivalence class, say
${\mathscr{V}}_{j}$ or ${\mathscr{C}}_{k}$. Assume that $x$ and
$y\in{\mathscr{V}}_{j}$. The argument is identical if we replace
${\mathscr{V}}_{j}$ by ${\mathscr{C}}_{k}$. Replace $\pi_{n}(x)$, $\pi_{n}(y)$
by $\pi_{n}(x)/\pi_{n}({\mathscr{V}}_{j})$,
$\pi_{n}(y)/\pi_{n}({\mathscr{V}}_{j})$, respectively. By Lemma 6.1,
$\pi_{n}(x)/\pi_{n}({\mathscr{V}}_{j})$ converges to
$\pi^{\sharp}_{j}(x)=\pi^{(1)}_{j}(x)>0$. Hence, (8.13) converges to

$\pi^{(1)}_{j}(x)\,{\mathbb{R}}_{0}(x,y)\,\Big{\\{}\sqrt{\frac{\mu(y)}{\pi^{(1)}_{j}(y)}}\,-\,\sqrt{\frac{\mu(x)}{\pi^{(1)}_{j}(x)}}\,\Big{\\}}^{2}\;.$

Putting together the previous estimates yields that
${\mathscr{I}}_{n}(\mu)\to{\mathscr{J}}^{(0)}(\mu)$. To complete the proof of
the $\Gamma-\limsup$, it remains to recall the statement of Lemma 8.1.

We turn to the $\Gamma-\liminf$. Fix $\mu\in{\mathscr{P}}(V)$ and a sequence
of probability measures $\mu_{n}$ in ${\mathscr{P}}({\mathscr{V}})$ converging
to $\mu$. By definition of ${\mathscr{I}}_{n}$,

${\mathscr{I}}_{n}(\mu_{n})\;\geq\;-\,\int_{V}\frac{{\mathscr{L}}_{n}u}{u}\,d\mu_{n}\;=\;-\,\sum_{x\in
V}\frac{\mu_{n}(x)}{u(x)}\,\sum_{y\in V}R_{n}(x,y)\,[\,u(y)-u(x)\,]$

for all $u:V\to(0,\infty)$. As $\mu_{n}\to\mu$ and $R_{n}\to{\mathbb{R}}_{0}$,
this expression converges to

$-\,\sum_{(x,y)\in{\mathbb{E}}_{0}}\frac{\mu(x)}{u(x)}\,{\mathbb{R}}_{0}(x,y)\,[\,u(y)-u(x)\,]\;.$

Therefore,

$\liminf_{n\to\infty}{\mathscr{I}}_{n}(\mu_{n})\;\geq\;\sup_{u>0}\,-\,\sum_{(x,y)\in{\mathbb{E}}_{0}}\frac{\mu(x)}{u(x)}\,{\mathbb{R}}_{0}(x,y)\,[\,u(y)-u(x)\,]\;=\;{\mathscr{I}}^{(0)}(\mu)\;,$

which completes the proof of the lemma. ∎

Recall from (8.9) the definition of the functionals ${\mathscr{I}}^{(p)}$,
$1\leq p\leq{\mathfrak{q}}$.

######  Proposition 8.4.

Fix $1\leq p\leq{\mathfrak{q}}$. The functional
$\theta^{(p)}_{n}\,{\mathscr{I}}_{n}$ $\Gamma$-converges to
${\mathscr{I}}^{(p)}$.

###### Proof.

We start with the $\Gamma-\limsup$. Fix $\mu\in{\mathscr{P}}(V)$. If $\mu$ is
not a convex combinations of the measures $\pi^{(p)}_{j}$, $j\in S_{p}$, there
is nothing to prove. Assume, therefore, that $\mu=\sum_{j\in
S_{p}}\omega_{j}\,\pi^{(p)}_{j}$ for some weights $\omega_{j}$.

Let $f_{n}:{\mathscr{V}}^{(p)}\to{\mathbb{R}}_{+}$ be the function given by
$f_{n}=\sum_{j\in S_{p}}\omega_{j}(n)\,\chi_{{}_{{\mathscr{V}}^{(p)}_{j}}}$,
where $\omega_{j}(n)=\omega_{j}/\pi_{n}({\mathscr{V}}^{(p)}_{j})$. To extend
this function to $V$, solve the Poisson equation (A.2) with
${\mathscr{L}}={\mathscr{L}}_{n}$, $V_{0}={\mathscr{V}}^{(p)}$,
$g=\sqrt{f_{n}}$. Denote by $h_{n}$ the solution of the equation. Let
$\mu_{n}=\alpha_{n}\,h^{2}_{n}\,\pi_{n}$, where $\alpha_{n}$ is a normalizing
constant which turns $\mu_{n}$ into a probability measure.

We claim that $\alpha_{n}\to 1$ and $\mu_{n}\to\mu$. By definition,

$\alpha_{n}^{-1}\;=\;\sum_{x\not\in{\mathscr{V}}^{(p)}}h_{n}(x)^{2}\,\pi_{n}(x)\;+\;\sum_{j\in
S_{p}}\sum_{x\in{\mathscr{V}}^{(p)}_{j}}f_{n}(x)\,\pi_{n}(x)\;.$

By definition of $h_{n}$, for $x\not\in{\mathscr{V}}^{(p)}$,

$\displaystyle h_{n}(x)^{2}\,\pi_{n}(x)$
$\displaystyle=\;\Big{\\{}\,\sum_{j\in
S_{p}}\sqrt{\omega_{j}(n)}\,{\mathbf{P}}^{n}_{\\!x}\big{[}\,H_{{\mathscr{V}}^{(p)}_{j}}=H_{{\mathscr{V}}^{(p)}}\,\big{]}\,\Big{\\}}^{2}\,\pi_{n}(x)$
$\displaystyle\leq\;C_{0}\,\sum_{j\in
S_{p}}\frac{\omega_{j}}{\pi_{n}({\mathscr{V}}^{(p)}_{j})}\,{\mathbf{P}}^{n}_{\\!x}\big{[}\,H_{{\mathscr{V}}^{(p)}_{j}}=H_{{\mathscr{V}}^{(p)}}\,\big{]}^{2}\,\pi_{n}(x)\;,$

where the constant $C_{0}$ bounds the cardinality of $V$. By Lemma 7.8, this
expression vanishes as $n\to\infty$. By definition of $f_{n}$, the second term
of the penultimate displayed equation is equal to

$\sum_{j\in
S_{p}}\sum_{x\in{\mathscr{V}}^{(p)}_{j}}\frac{\omega_{j}}{\pi_{n}({\mathscr{V}}^{(p)}_{j})}\,\pi_{n}(x)\;=\;1\;,$

which proves that $\alpha_{n}\to 1$.

The previous argument shows that
$\mu_{n}(x)\,=\,\alpha_{n}\,h_{n}(x)^{2}\,\pi_{n}(x)\to 0=\mu(x)$ if
$x\not\in{\mathscr{V}}^{(p)}$. If $x\in{\mathscr{V}}^{(p)}_{j}$,
$\mu_{n}(x)\,=\,\alpha_{n}\,f_{n}(x)\,\pi_{n}(x)\,=\,\alpha_{n}\,\omega_{j}\,\pi_{n}(x)/\pi_{n}({\mathscr{V}}^{(p)}_{j})$.
Since $\alpha_{n}\to 1$, by Corollary 6.3, the previous expression converges
to $\omega_{j}\,\pi^{(p)}_{j}(x)\,=\,\mu(x)$.

To complete the proof of the $\Gamma-\limsup$, it remains to show that
$\limsup_{n}\theta^{(p)}_{n}\,{\mathscr{I}}_{n}(\mu_{n})\leq{\mathscr{I}}^{(p)}(\mu)$.
By (3.10) and the definition of $\mu_{n}$,

${\mathscr{I}}_{n}(\mu_{n})\;=\;\alpha_{n}\,\langle\,h_{n}\,,\,(-{\mathscr{L}}_{n})\,h_{n}\,\rangle_{\pi_{n}}\;.$

By Corollary A.2 and the definition of $h_{n}$, the right-hand side is equal
to

$\displaystyle\alpha_{n}\,\pi_{n}({\mathscr{V}}^{(p)})\,\langle\sqrt{f_{n}}\,,(-{\mathscr{L}}^{(p)}_{n})\sqrt{f_{n}}\rangle_{\pi^{(p)}_{n}}$
$\displaystyle\quad=\;-\,\alpha_{n}\,\sum_{x,y\in{\mathscr{V}}^{(p)}}\pi_{n}(x)\,R^{(p)}_{n}(x,y)\,\sqrt{f_{n}(x)}\,\big{\\{}\,\sqrt{f_{n}(y)}\,-\,\sqrt{f_{n}(x)}\,\big{\\}}\;,$

where
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathscr{L}}^{(p)}_{n}$
stands for the generator of the trace process $Y^{n,p}_{t}$ introduced in
(2.8), and
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pi^{(p)}_{n}$
for the measure $\pi_{n}$ conditioned to ${\mathscr{V}}^{(p)}$. Since $f_{n}$
is constant and equal to $\omega_{j}(n)$ on each set
${\mathscr{V}}^{(p)}_{j}$, the previous expression is equal to

$\displaystyle-\,\alpha_{n}\,\sum_{j\in S_{p}}\sqrt{\omega_{j}(n)}\sum_{k\in
S_{p}\setminus\\{j\\}}\,\big{\\{}\,\sqrt{\omega_{k}(n)}\,-\,\sqrt{\omega_{j}(n)}\,\big{\\}}\sum_{x\in{\mathscr{V}}^{(p)}_{j}}\pi_{n}(x)\,\sum_{y\in{\mathscr{V}}^{(p)}_{k}}R^{(p)}_{n}(x,y)$
$\displaystyle\quad=\;-\,\alpha_{n}\,\sum_{j\in
S_{p}}\sqrt{\omega_{j}(n)}\sum_{k\in
S_{p}\setminus\\{j\\}}\,\big{\\{}\,\sqrt{\omega_{k}(n)}\,-\,\sqrt{\omega_{j}(n)}\,\big{\\}}\pi_{n}({\mathscr{V}}^{(p)}_{j})\,r^{(p)}_{n}(j,k)\;,$

where $r^{(p)}_{n}(j,k)$ is defined in (2.9). Up to this point, we proved that

$\theta^{(p)}_{n}\,{\mathscr{I}}_{n}(\mu_{n})\;=\;\frac{\alpha_{n}}{2}\,\theta^{(p)}_{n}\,\sum_{j,k\in
S_{p}}\pi_{n}({\mathscr{V}}^{(p)}_{j})\,r^{(p)}_{n}(j,k)\,\Big{\\{}\,\sqrt{\omega_{k}(n)}\,-\,\sqrt{\omega_{j}(n)}\,\Big{\\}}^{2}\;,$

where we used that
$\pi_{n}({\mathscr{V}}^{(p)}_{j})\,r^{(p)}_{n}(j,k)\,=\,\pi_{n}({\mathscr{V}}^{(p)}_{k})\,r^{(p)}_{n}(k,j)$,
an identity which follows from the reversibility assumption.

Recall that $\alpha_{n}\to 1$. In view of the definition of $\omega_{i}(n)$,
it remains to examine the asymptotic behavior of

$\pi_{n}({\mathscr{V}}^{(p)}_{j})\,\theta^{(p)}_{n}\,r^{(p)}_{n}(j,k)\,\Big{\\{}\,\sqrt{\frac{\omega_{k}}{\pi_{n}({\mathscr{V}}^{(p)}_{k})}}\,-\,\sqrt{\frac{\omega_{j}}{\pi_{n}({\mathscr{V}}^{(p)}_{j})}}\,\Big{\\}}^{2}\;.$
(8.14)

As in the proof of Proposition 8.3, we divide the pairs $(j,k)$ in three
types. Assume first that $\theta^{(p)}_{n}\,r^{(p)}_{n}(j,k)\to 0$ and
$\theta^{(p)}_{n}\,r^{(p)}_{n}(k,j)\to 0$. By [43, Lemma 3.1], and (2.16),
either $\pi_{n}({\mathscr{V}}^{(p)}_{j})/\pi_{n}({\mathscr{V}}^{(p)}_{k})$
converges to a nonnegative real number or so does
$\pi_{n}({\mathscr{V}}^{(p)}_{k})/\pi_{n}({\mathscr{V}}^{(p)}_{j})$. Assume
that $\pi_{n}({\mathscr{V}}^{(p)}_{k})/\pi_{n}({\mathscr{V}}^{(p)}_{j})\to
a\in[0,\infty)$. In this case, by reversibility, (8.14) is equal to

$\theta^{(p)}_{n}\,r^{(p)}_{n}(k,j)\,\Big{\\{}\,\sqrt{\omega_{k}}\,-\,\sqrt{\omega_{j}\,\frac{\pi_{n}({\mathscr{V}}^{(p)}_{k})}{\pi_{n}({\mathscr{V}}^{(p)}_{j})}}\,\Big{\\}}^{2}\;,$
(8.15)

which vanishes as $n\to\infty$.

Next, suppose that $\theta^{(p)}_{n}\,r^{(p)}_{n}(j,k)\to 0$ and
$\theta^{(p)}_{n}\,r^{(p)}_{n}(k,j)\to r^{(p)}(k,j)>0$, where $r^{(p)}(k,j)$
has been introduced in (2.10). In particular, $k$ is a transient state of the
chain ${\mathbb{X}}^{(p)}_{t}$. By reversibility,
$\pi_{n}({\mathscr{V}}^{(p)}_{k})/\pi_{n}({\mathscr{V}}^{(p)}_{j})\,=\,r^{(p)}_{n}(j,k)/r^{(p)}_{n}(k,j)\to
0$. Hence, (8.14), which is equal to (8.15), converges to

$r^{(p)}(k,j)\,\omega_{k}\;.$

Finally, suppose that $\theta^{(p)}_{n}\,r^{(p)}_{n}(j,k)\to r^{(p)}(j,k)>0$
and $\theta^{(p)}_{n}\,r^{(p)}_{n}(k,j)\to r^{(p)}(k,j)>0$. This means that
$j$ and $k$ belong to some closed irreducible class ${\mathfrak{R}}^{(p)}_{m}$
of the chain ${\mathbb{X}}^{(p)}_{t}$. By Lemma 6.2, the expression (8.14)
converges to

$M^{(p)}_{m}(j)\,r^{(p)}(k,j)\,\Big{\\{}\,\sqrt{\frac{\omega_{k}}{M^{(p)}_{m}(k)}}\,-\,\sqrt{\frac{\omega_{j}}{M^{(p)}_{m}(j)}}\,\Big{\\}}^{2}\;.$

Combining the previous estimates yields that
$\theta^{(p)}_{n}{\mathscr{I}}_{n}(\mu_{n})$ converges to
${\mathscr{J}}^{(p)}(\mu)$, which completes the proof of the $\Gamma-\limsup$
in view of Lemma 8.2.

We turn to the $\Gamma-\liminf$ where we use an induction argument. Fix $1\leq
p\leq{\mathfrak{q}}$ and assume that the $\Gamma$-convergence of
$\theta^{(p-1)}_{n}\,{\mathscr{I}}_{n}$ to ${\mathscr{I}}^{(p-1)}$ has been
proved. Fix a probability measure $\mu$ on $V$ and a sequence $\mu_{n}$
converging to $\mu$.

Suppose that ${\mathscr{I}}^{(p-1)}(\mu)>0$. In this case, since
$\theta^{(p-1)}_{n}\,{\mathscr{I}}_{n}$ $\Gamma$-converges to
${\mathscr{I}}^{(p-1)}$ and $\theta^{(p)}_{n}/\theta^{(p-1)}_{n}\to\infty$,

$\liminf_{n\to\infty}\theta^{(p)}_{n}\,{\mathscr{I}}_{n}(\mu_{n})\;=\;\liminf_{n\to\infty}\frac{\theta^{(p)}_{n}}{\theta^{(p-1)}_{n}}\,\theta^{(p-1)}_{n}\,{\mathscr{I}}_{n}(\mu_{n})\;\geq\;{\mathscr{I}}^{(p-1)}(\mu)\,\lim_{n\to\infty}\frac{\theta^{(p)}_{n}}{\theta^{(p-1)}_{n}}\;=\;\infty\;.$

On the other hand, by (8.12), ${\mathscr{I}}^{(p-1)}(\mu)=\infty$. This proves
the $\Gamma-\liminf$ convergence for measures $\mu$ such that
${\mathscr{I}}^{(p-1)}(\mu)>0$.

Assume that ${\mathscr{I}}^{(p-1)}(\mu)=0$. By (8.10), there exists a
probability measure $\omega$ on $S_{p}$ such that $\mu=\sum_{j\in
S_{p}}\omega_{j}\,\pi^{(p)}_{j}$. By (3.9),

${\mathscr{I}}_{n}(\mu_{n})\;\geq\;-\,\int_{V}\frac{{\mathscr{L}}_{n}u}{u}\,d\mu_{n}$

for all $u:V\to(0,\infty)$.

Fix a function $h:{\mathscr{V}}^{(p)}\to(0,\infty)$ which is constant on each
${\mathscr{V}}^{(p)}_{j}$, $j\in S_{p}$: $h=\sum_{j\in
S_{p}}{\mathbf{h}}(j)\,\chi_{{\mathscr{V}}^{(p)}_{j}}$. Let $u_{n}\colon
V\to{\mathbb{R}}$ be the solution of the Poisson equation (A.2) with
${\mathscr{L}}={\mathscr{L}}_{n}$, $V_{0}={\mathscr{V}}^{(p)}$ and $g=h$. By
the representation (A.3), it is clear that $u_{n}(x)\in(0,\infty)$ for all
$x\in V$.

Since $u_{n}$ is harmonic on $V\setminus{\mathscr{V}}^{(p)}$ and $u_{n}=h$ on
${\mathscr{V}}^{(p)}$, by Lemma A.1, the right-hand side of the previous
displayed equation with $u=u_{n}$ is equal to

$-\,\int_{{\mathscr{V}}^{(p)}}\frac{{\mathscr{L}}_{n}u_{n}}{u_{n}}\,d\mu_{n}\;=\;-\,\int_{{\mathscr{V}}^{(p)}}\frac{{\mathscr{L}}_{n}u_{n}}{h}\,d\mu_{n}\;=\;-\,\int_{{\mathscr{V}}^{(p)}}\frac{{\mathscr{L}}^{(p)}_{n}h}{h}\,d\mu_{n}\;.$

Here, as in the first part of the proof, ${\mathscr{L}}^{(p)}_{n}$ stands for
the generator of the trace process $Y^{n,p}_{t}$ introduced in (2.8).

Since $h$ is constant on each set ${\mathscr{V}}^{(p)}_{j}$ (and equal to
${\mathbf{h}}(j)$), the last integral is equal to

$-\,\sum_{j,k\in
S_{p}}\frac{[\,{\mathbf{h}}(k)-{\mathbf{h}}(j)\,]}{{\mathbf{h}}(j)}\,\sum_{x\in{\mathscr{V}}^{(p)}_{j}}\pi_{n}(x)\,\frac{\mu_{n}(x)}{\pi_{n}(x)}\,R^{(p)}_{n}(x,{\mathscr{V}}^{(p)}_{k})\;,$

where
$R^{(p)}_{n}(x,{\mathscr{V}}^{(p)}_{k})=\sum_{y\in{\mathscr{V}}^{(p)}_{k}}R^{(p)}_{n}(x,y)$.
By Proposition 3.2,
$\pi_{n}(x)/\pi_{n}({\mathscr{V}}^{(p)}_{j})\to\pi^{(p)}_{j}(x)$ for all
$x\in{\mathscr{V}}^{(p)}_{j}$. Thus, since $\mu_{n}\to\mu=\sum_{j\in
S_{p}}\omega_{j}\,\pi^{(p)}_{j}$,

$\lim_{n\to\infty}\pi_{n}({\mathscr{V}}^{(p)}_{j})\,\frac{\mu_{n}(x)}{\pi_{n}(x)}\;=\;\omega_{j}\quad\text{for
all $x\in{\mathscr{V}}^{(p)}_{j}$}\;.$

Therefore, by (2.10), as $n\to\infty$, the penultimate expression multiplied
by $\theta^{(p)}_{n}$ converges to

$-\,\sum_{j\in S_{p}}\omega_{j}\,\frac{1}{{\mathbf{h}}(j)}\,\sum_{k\in
S_{p}}r^{(p)}(j,k)\,[\,{\mathbf{h}}(k)-{\mathbf{h}}(j)\,]\;=\;-\,\sum_{j\in
S_{p}}\omega_{j}\,\frac{{\mathbb{L}}^{(p)}{\mathbf{h}}}{{\mathbf{h}}}\;.$

Summarising, we proved that

$\liminf_{n\to\infty}\theta^{(p)}_{n}\,{\mathscr{I}}_{n}(\mu_{n})\;\geq\;\sup_{{\mathbf{h}}}\,-\,\sum_{j\in
S_{p}}\omega_{j}\,\frac{{\mathbb{L}}^{(p)}{\mathbf{h}}}{{\mathbf{h}}}\;,$

where the supremum is carried over all functions
${\mathbf{h}}:S_{p}\to(0,\infty)$. By (3.13), the right-hand side is precisely
${\mathscr{I}}^{(p)}(\mu)$, which completes the proof of the $\Gamma-\liminf$.
∎

##  Appendix A Potential theory

We present in this section some results on potential theory used in the
article. We do not assume reversibility. We keep the same notation of the
article, removing the index $n$. In particular, $X_{t}$ is a $V$-valued,
continuous-time irreducible Markov process whose jump rates are represented by
$R(x,y)$. Denote by
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}({\mathcal{F}}_{t}:t\geq
0)$ the canonical filtration induced by the chain $X_{t}$. Hence,
${\mathcal{F}}_{t}$ is the $\sigma$-algebra generated by the variables
$X_{s}$, $0\leq s\leq t$.

We first recall for the reader’s convenience the definition of the trace of a
process on a subset.

### Trace process

Fix a non-empty subset $W$ of $V$. Denote by $T^{W}(t)$ the total time the
process $X_{t}$ spends in $W$ in the time-interval $[0,t]$:

$T^{W}(t)\;=\;\int_{0}^{t}\,\chi_{W}(X_{s})\,ds\;,$

where, recall, $\chi_{W}$ represents the indicator function of the set $W$.
Denote by $S^{W}(t)$ the generalized inverse of $T^{W}(t)$:

$S^{W}(t)\;=\;\sup\\{\,s\geq 0\,:\,T^{W}(s)\leq t\,\\}\;.$

The trace of $X_{t}$ on $W$, denoted by
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(X^{W}_{t}:t\geq
0)$, is defined by

$X^{W}_{t}\;=\;X_{S^{W}(t)}\;;\;\;\;t\geq 0\;.$ (A.1)

By Propositions 6.1 and 6.3 in [1], the trace process is an irreducible,
$W$-valued continuous-time Markov chain, obtained by turning off the clock
when the process $X_{t}$ visits the set $W^{c}$, that is, by deleting all
excursions to $W^{c}$. For this reason, it is called the trace process of
$X_{t}$ on $W$.

Denote by
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathscr{L}}_{W}$,
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}R_{W}$,
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\lambda_{W}$,
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}p_{W}$ and
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pi_{W}$ its
generator, jump rates, holding times, transition matrix and stationary state,
respectively. The measure $\pi_{W}$ is obtained by conditioning $\pi$ to $W$:
$\pi_{W}(x)=\pi(x)/\pi(W)$.

Let
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathbf{P}}^{W}_{\\!x}$,
$x\in W$, be the probability measure on the path space $D({\mathbb{R}}_{+},W)$
induced by the Markov chain $X^{W}_{t}$ starting from $x$. Expectation with
respect to ${\mathbf{P}}^{W}_{x}$ is represented by
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathbf{E}}^{W}_{x}$.

### Poisson equation

Fix a non-empty proper subset $V_{0}$ of $V$ and a function
$g:V_{0}\to{\mathbb{R}}$. Let $f$ be the solution of the Poisson equation

$\left\\{\begin{aligned} &{\mathscr{L}}f\,=\,0\;,\quad V\setminus V_{0}\;,\\\
&f\,=\,g\;,\quad V_{0}\;.\end{aligned}\right.$ (A.2)

Recall from (2.5) the definition of the hitting and return times to a subset
${\mathscr{A}}$ of $V$. By the strong Markov property, the solution of the
Poisson equation can be represented as

$f(x)\,=\,\;{\mathbf{E}}_{x}[\,g(X_{H_{V_{0}}})\,]\;,\quad x\in V\;.$ (A.3)

Fix $V_{0}\subset W\subset V$ and denote by $f_{W}$ the solution of the
Poisson equation

$\left\\{\begin{aligned} &{\mathscr{L}}_{W}f\,=\,0\;,\quad W\setminus
V_{0}\;,\\\ &f\,=\,g\;,\quad V_{0}\;.\end{aligned}\right.$ (A.4)

Mind that $W$ may be equal to $V_{0}$.

Starting from $x\not\in V_{0}$, the processes $X_{t}$ and $X^{W}_{t}$ hit the
set $V_{0}$ at the same point: $X^{W}_{H_{V_{0}}(X^{W})}=X_{H_{V_{0}}}$,
${\mathbf{P}}_{\\!x}$ a.s. In this formula and below,
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}H_{V_{0}}(X^{W})$,
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}H^{+}_{V_{0}}(X^{W})$
stand for hitting and return time to $V_{0}$ for the process $X^{W}$. By the
representation (A.3) and the previous observation, for $x\in W$

$f_{W}(x)\;=\;{\mathbf{E}}^{W}_{x}[\,g(X_{H_{V_{0}}})\,]\;=\;{\mathbf{E}}_{x}[\,g(X^{W}_{H_{V_{0}}(X^{W})})\,]\;=\;{\mathbf{E}}_{x}[\,g(X_{H_{V_{0}}})\,]\;=\;f(x)\;.$
(A.5)

######  Lemma A.1.

Fix $V_{0}\subset W\subset V$. Denote by $f$, $f_{W}$ the solutions of the
Poisson equations (A.2), (A.4), respectively. Then,

$({\mathscr{L}}f)(x)\;=\;({\mathscr{L}}_{W}f_{W})(x)\;,\quad x\in V_{0}\;.$

###### Proof.

Fix $x\in V_{0}$. The left-hand side of the identity appearing in the
statement of the lemma can be written as

$\lambda(x)\,\sum_{y\in V}p(x,y)\,[\,f(y)\,-\,f(x)\,]\;.$

Without loss of generality, assume that $p(z,z)=0$ for all $z\in V$ (if this
is not the case, one redefines the holding time $\lambda(z)$ for the identity
to hold). By (A.3), $f(y)={\mathbf{E}}_{y}[\,g(X_{H_{V_{0}}})\,]$ for all
$y\in V$, and by the strong Markov property
${\mathbf{E}}_{x}[\,g(X_{H^{+}_{V_{0}}})\,]=\sum_{y\in V}p(x,y)\,f(y)$. Hence,
the previous sum can be written as

$\lambda(x)\,\,\big{\\{}\,{\mathbf{E}}_{x}[\,g(X_{H^{+}_{V_{0}}})\,]\,-\,f(x)\,\big{\\}}\;.$
(A.6)

Recall that we denote by $X^{W}_{t}$ the trace of the process $X_{t}$ on $W$.
We consider two cases. If $H^{+}_{W}<H^{+}_{x}$ then the process $X_{t}$ and
$X^{W}_{t}$ return to $V_{0}$ at the same point $X_{H^{+}_{V_{0}}}$ (to prove
this assertion, consider separately the two situations
$\\{H^{+}_{V_{0}}<H^{+}_{x}\\}$ and $\\{H^{+}_{V_{0}}=H^{+}_{x}\\}$). Thus, if
$H^{+}_{W}<H^{+}_{x}$ we may replace in (A.6) $g(X_{H^{+}_{V_{0}}})$ by
$g(X^{W}_{H^{+}_{V_{0}}(X^{W})})$.

If $H^{+}_{W}=H^{+}_{x}$, the process $X_{t}$ returns to $V_{0}$ (and also to
$W$) at $x$. In contrast, in the time interval $[0,H^{+}_{W}]$ the trace
process on $W$ remains at $x$, and $X^{W}_{t}$ may return to $V_{0}$ at a
point $y\neq x$. In particular, $X_{t}$ and $X^{W}_{t}$ may return to $V$ at
different points. In this case, since $H^{+}_{V_{0}}=H^{+}_{x}$, we have

${\mathbf{E}}_{x}\big{[}\,g(X_{H^{+}_{V_{0}}})\,\chi(H^{+}_{x}=H^{+}_{W})\,\big{]}\;=\;g(x)\,{\mathbf{P}}_{x}\big{[}\,H^{+}_{x}=H^{+}_{W}\,\big{]}\;.$

Up to this point, we proved that for $x\in V_{0}$,

${\mathbf{E}}_{x}\big{[}\,g(X_{H^{+}_{V_{0}}})\,\big{]}\;=\;{\mathbf{E}}_{x}\big{[}\,g(X^{W}_{H^{+}_{V_{0}}(X^{W})})\,\chi_{{}_{A^{c}}}\,\big{]}\;+\;g(x)\,{\mathbf{P}}_{x}[\,A\,]\;,$

where $A$ is the event $\\{H^{+}_{x}=H^{+}_{W}\\}$. Write $\chi_{{}_{A^{c}}}$
as $1-\chi_{{}_{A}}$. On the event $A$,
$H^{+}_{V_{0}}(X^{W})=H^{+}_{V_{0}}+H^{+}_{V_{0}}(X^{W})\circ\vartheta_{H^{+}_{V_{0}}}$.
Hence, conditioning on ${\mathcal{F}}_{H^{+}_{V_{0}}}$, since $A$ is
${\mathcal{F}}_{H^{+}_{V_{0}}}$-measurable and $X(H^{+}_{V_{0}})=x$ on the
event $A$, by the strong Markov property,

${\mathbf{E}}_{x}\big{[}\,g(X^{W}_{H^{+}_{V_{0}}(X^{W})})\,\chi_{{}_{A}}\,\big{]}\;=\;{\mathbf{P}}_{x}[\,A\,]\,{\mathbf{E}}_{x}\big{[}\,g(X^{W}_{H^{+}_{V_{0}}(X^{W})})\,\big{]}\;.$

Therefore, for $x\in V_{0}$,

${\mathbf{E}}_{x}\big{[}\,g(X_{H^{+}_{V_{0}}})\,\big{]}\;-\;g(x)\;=\;{\mathbf{P}}_{x}[\,H^{+}_{W}\,<\,H^{+}_{x}\,]\;\Big{\\{}\,{\mathbf{E}}_{x}\big{[}\,g(X^{W}_{H^{+}_{V_{0}}(X^{W})})\,\big{]}\;-\;g(x)\,\Big{\\}}\;.$
(A.7)

By equation (6.9) in [1],
$\lambda(x)\,{\mathbf{P}}_{x}[\,H^{+}_{W}\,<\,H^{+}_{x}\,]=\lambda_{W}(x)$.
Therefore, (A.6) is equal to

$\lambda_{W}(x)\,\,\big{\\{}\,{\mathbf{E}}^{W}_{x}[\,g(X_{H^{+}_{V_{0}}})\,]\,-\,f(x)\,\big{\\}}\;.$

By the strong Markov property and (A.5), this expression is equal to

$\displaystyle\lambda_{W}(x)\,\sum_{y\in
W}p_{W}(x,y)\,\big{\\{}\,{\mathbf{E}}^{W}_{y}[\,g(X_{H_{V_{0}}})\,]\,-\,f(x)\,\big{\\}}$
$\displaystyle\quad\;=\;\lambda_{W}(x)\,\sum_{y\in
W}p_{W}(x,y)\,\big{\\{}\,f_{W}(y)\,-\,f_{W}(x)\,\big{\\}}\;=\;({\mathscr{L}}_{W}f_{W})(x)\;,$

as claimed. ∎

Denote by $D(f)$ the Dirichlet form of a function $f:V\to{\mathbb{R}}$:

$D(f)\;:=\;\langle\,f\,,\,(-\,{\mathscr{L}})f\,\rangle_{\pi}\;.$

######  Corollary A.2.

Fix $V_{0}\subset W\subset V$. Denote by $f$, $f_{W}$ the solutions of the
Poisson equations (A.2), (A.4), respectively. Then,

$D(f)\;=\;\pi(W)\,\langle\,f_{W}\,,\,(-\,{\mathscr{L}}_{W})f_{W}\,\rangle_{\pi_{W}}\;.$
(A.8)

###### Proof.

By definition of the Dirichlet form,

$D(f)\;=\;\langle\,f\,,\,(-\,{\mathscr{L}})f\,\rangle_{\pi}\;=\;-\,\sum_{x\in
V}\pi(x)\,f(x)\,({\mathscr{L}}f)(x)\;.$

Since $f$ is harmonic on $V_{0}^{c}$, the sum can be restricted to $V_{0}$.
Hence, the previous expression is equal to

$-\,\sum_{x\in V_{0}}\pi(x)\,f(x)\,({\mathscr{L}}f)(x)\;.$

By (A.5) and Lemma A.1, this sum is equal to

$-\,\sum_{x\in V_{0}}\pi(x)\,f_{W}(x)\,({\mathscr{L}}_{W}f_{W})(x)\;.$

Since $f_{W}$ is ${\mathscr{L}}_{W}$-harmonic on $W\setminus V_{0}$, we may
extend the sum to $W$. To complete the proof, it remains to recall that
$\pi_{W}(\,\cdot\,)=\pi(\,\cdot\,)/\pi(W)$. ∎

The same proof yields the following result.

######  Corollary A.3.

Fix $V_{0}\subset V$, $g:V_{0}\to{\mathbb{R}}$, and let $u$ be the solution of
(A.2). Then,

$\int_{V}\frac{{\mathscr{L}}\,u}{u}\,d\mu\;=\;\int_{V_{0}}\frac{{\mathscr{L}}_{V_{0}}\,g}{g}\,d\mu$

for all probability measures $\mu$ on $V$.

###### Proof.

Since $u$ is harmonic on $V\setminus V_{0}$, we may restrict the integral to
$V_{0}$. By Lemma A.1, on $V_{0}$ we may replace ${\mathscr{L}}u$ by
${\mathscr{L}}_{V_{0}}u_{V_{0}}$, where $u_{V_{0}}$ is the solution of (A.4)
with $W=V_{0}$. However, as $W=V_{0}$, the solution of (A.4) is $u_{V_{0}}=g$.
Hence, ${\mathscr{L}}_{V_{0}}u_{V_{0}}={\mathscr{L}}_{V_{0}}g$. As $u=g$ on
$V_{0}$, the proof is complete. ∎

We turn to an estimate of hitting times. Denote by $\pi_{A}$, $A\subset V$,
the stationary measure $\pi$ conditioned to $A$

$\pi_{A}(x)\;=\;\frac{\pi(x)}{\pi(A)}\;,\quad x\in A\;.$

Next result is [36, Proposition 8.4]. It holds for non-reversible dynamics.
The assertion in the case where $A$ is a singleton follows from the proofs of
[5, Corollary 4.2] and [36, Proposition 8.4].

######  Lemma A.4.

Let $A$, $B$ be two nonempty disjoint subsets of $E$. Then, for every
probability measure $\nu$ concentrated on the set $A$ and $\varrho>0$

$\mathbb{\mathbf{P}}_{\nu}\big{[}\,H_{B}\;\leq\;\varrho\,\big{]}^{2}\;\leq\;e^{2}\,{E}_{\pi_{A}}\Big{[}\,\Big{(}\frac{\nu}{\pi_{A}}\Big{)}^{2}\,\Big{]}\,\frac{{\rm
cap}(A,\,B)}{\pi(A)}\,\varrho\;.$

If $A$ is a singleton, $A=\\{x\\}$, then for every $\varrho>0$

$\mathbb{\mathbf{P}}_{\\!x}\big{[}\,H_{B}\;\leq\;\varrho\,\big{]}\;\leq\;e\,\frac{{\rm
cap}(\\{x\\},\,B)}{\pi(x)}\,\varrho\;.$

This result helps in showing that the left-hand side vanishes asymptotically
if $[\,{\rm cap}_{n}(\\{x\\},\,B)/\pi_{n}(x)\,]\,\varrho_{n}\to 0$.

######  Remark A.5.

For two sets $A$, $B$ satisfying the hypotheses of Lemma A.4, let $\nu_{A,B}$
be the equilibrium measure on $A$:

$\nu_{A,B}(x)\;=\;\frac{1}{{\rm
cap}(A\,,\,B)}\,\pi(x)\,\lambda(x)\,\mathbb{\mathbf{P}}_{\\!x}[\,H_{B}<H^{+}_{A}\,]\;,\quad
x\in A\;.$

By Chebychev inequality and [4, Proposition A.2],

$\mathbb{\mathbf{P}}_{\\!\nu_{A,B}}\big{[}\,H_{B}\;\geq\;\varrho\,\big{]}\;\leq\;\frac{1}{\varrho}\,E_{\nu_{A,B}}\big{[}\,H_{B}\,\big{]}\;=\;\frac{E_{\pi}[h^{*}_{A,B}]}{\varrho\,{\rm
cap}(A,\,B)}\;,$

where $h^{*}_{A,B}$ stands for the equilibrium potential of the time-reversed
process (sometimes called the adjoint process):
$h^{*}_{A,B}(y)={\mathbf{P}}^{*}_{\\!y}[H_{A}<H_{B}]$, and ${\mathbf{P}}^{*}$
stands for the distribution of the continuous-time Markov chain with jump
rates $R^{*}(x,y)$ given by $R^{*}(x,y)=\pi(y)\,R(y,x)/\pi(x)$. In many cases,
$E_{\nu_{A,B}}[\,H_{B}\,]\,=\,[1+o(1)]\,\pi(A)$ so that

$\mathbb{\mathbf{P}}_{\\!\nu_{A,B}}\big{[}\,H_{B}\;\geq\;\varrho\,\big{]}\;\leq\;[1+o(1)]\,\frac{\pi(A)}{\varrho\,{\rm
cap}(A,\,B)}\;.$

This inequality demonstrates that the bound in Lemma A.4 is sharp whenever
$E_{\nu_{A,B}}[\,H_{B}\,]\,=\,[1+o(1)]\,\pi(A)$.

### Acknowledgments

C. L. has been partially supported by FAPERJ CNE E-26/201.207/2014, by CNPq
Bolsa de Produtividade em Pesquisa PQ 303538/2014-7.

## References

  * [1] J. Beltrán, C. Landim: Tunneling and metastability of continuous time Markov chains. J. Stat. Phys. 140 1065–1114 (2010).
  * [2] J. Beltrán, C. Landim: Metastability of reversible condensed zero range processes on a finite set, Probab. Theory Relat. Fields 152, 781–807 (2012).
  * [3] J. Beltrán, C. Landim; Metastability of reversible finite state Markov processes. Stoch. Proc. Appl. 121 1633–1677 (2011).
  * [4] J. Beltrán, C. Landim; Tunneling and metastability of continuous time Markov chains II. J. Stat. Phys. 149, 598–618 (2012).
  * [5] J. Beltrán, C. Landim: Martingale approach to metastability. Probab. Theory Relat. Fields 161, 267–307 (2015)
  * [6] J. Beltrán, C. Landim; Tunneling of the Kawasaki dynamics at low temperatures in two dimensions. Ann. Inst. H. Poincaré, Probab. Statist. 51, 59–88 (2015).
  * [7] G. Ben Arous, R. Cerf, Metastability of the three-dimensional Ising model on a torus at very low temperature, Electron. J. Probab. 1 (1996) Research Paper 10.
  * [8] L. Bertini, A. Faggionato, D. Gabrielli: Large deviations of the empirical flow for continuous time Markov chains. Annales de l’I.H.P. Probabilités et statistiques, 51, 867–900 (2015).
  * [9] A. Bianchi, S. Dommers, C. Giardinà: Metastability in the reversible inclusion process. Electron. J. Probab. 22, paper no. 70, (2017).
  * [10] A. Bianchi, A. Bovier, D. Ioffe: Sharp asymptotics for metastability in the random field Curie-Weiss model Electron. J. Probab. 14: Paper no. 53, pages 1541–1603, (2009).
  * [11] A. Bovier, M. Eckhoff, V. Gayrard, M. Klein: Metastability in stochastic dynamics of disordered mean field models. Probab. Theory Relat. Fields 119, 99-161 (2001).
  * [12] A. Bovier, M. Eckhoff, V. Gayrard, M. Klein: Metastability in reversible diffusion process I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. 6, 399–424 (2004)
  * [13] A. Bovier, F. den Hollander: Metastability: a potential-theoretic approach. Grundlehren der mathematischen Wissenschaften 351, Springer, Berlin, 2015.
  * [14] A. Bovier, F. den Hollander, F.R. Nardi: Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary. Probab. Theory Relat. Fields 135, 265–310 (2006).
  * [15] A. Bovier, F. Manzo: Metastability in Glauber dynamics in the low-temperature limit: beyond exponential asymptotics, J. Stat. Phys. 107, 757–779 (2002).
  * [16] A. Braides; _$\Gamma$ -Convergence for beginners._ Oxford University Press, Oxford, 2002.
  * [17] J. Cao, P. Chleboun, S. Grosskinsky: Dynamics of condensation in the totally asymmetric inclusion process. J. Stat. Phys. 155, 523–543 (2014).
  * [18] E. Cirillo, E. Olivieri: Metastability and nucleation for the Blume–Capel model. Different mechanisms of transition. J. Stat. Phys.83, 473-554 (1996)
  * [19] M. D. Donsker, S. R. S. Varadhan: Asymptotic evaluation of certain Markov proces expectations for large time, I, Comm. Pure Appl. Math., 28, 1-47 (1975).
  * [20] M. Freidlin, L. Koralov: Metastable Distributions of Markov Chains with Rare Transitions. J. Stat. Phys. 167, 1355–1375 (2017)
  * [21] M. I. Freidlin, A. D. Wentzell: Random perturbations of dynamical systems. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260. Springer-Verlag, New York, 1998.
  * [22] A. Gaudillière, F. Den Hollander, F.R. Nardi, E. Olivieri, E. Scoppola: Ideal gas approximation for a two-dimensional rarefied gas under Kawasaki dynamics, Stochastic Process. Appl. 119, 737–774 (2009).
  * [23] G. Di Gesù, M. Mariani: Full metastable asymptotic of the Fisher information. SIAM J. MATH. ANAL. 49, 3048–3072 (2017)
  * [24] B. Gois, C. Landim; Zero-temperature limit of the Kawasaki dynamics for the Ising lattice gas in a large two-dimensional torus. Ann. Probab. 43, 2151 – 2203 (2015).
  * [25] S. Grosskinsky, F. Redig, K. Vafayi: Dynamics of condensation in the symmetric inclusion process 18, article no. 66 (2013).
  * [26] F. den Hollander, F. Nardi, A. Troiani: Metastability for Kawasaki dynamics at low temperature with two types of particles Electron. J. Probab. 17: 1–26 (2012).
  * [27] S. Kim. Second time scale of the metastability of reversible inclusion processes. Probab. Theory Relat. Fields 180, 1135–1187 (2021).
  * [28] S. Kim, I. Seo: Condensation and Metastable Behavior of Non-reversible Inclusion Processes. Commun. Math. Phys. 382, 1343–1401 (2021).
  * [29] S. Kim, I. Seo: Metastability of Stochastic Ising and Potts Models on Lattices without External Fields. (2021) arXiv:2102.05565
  * [30] S. Kim, I. Seo: Metastability of Ising and Potts models without external fields in large volumes at low temperatures. arXiv:2109.13583 (2021)
  * [31] C. Landim: Metastability for a non-reversible dynamics: the evolution of the condensate in totally asymmetric zero range processes. Commun. Math. Phys. 330, 1–32 (2014).
  * [32] C. Landim: Metastable Markov chains. Probability Surveys 16, 143–227 (2019). DOI: 10.1214/18-PS310
  * [33] C. Landim, P. Lemire; Metastability of the two-dimensional Blume-Capel model with zero chemical potential and small magnetic field. J. Stat. Phys. 164, 346–376 (2016)
  * [34] C. Landim, P. Lemire, M. Mourragui: Metastability of the two-dimensional Blume-Capel model with zero chemical potential and small magnetic field on a large torus. J. Stat. Phys. 175, 456–494 (2019)
  * [35] C. Landim, M. Loulakis, M. Mourragui: Metastable Markov chains: from the convergence of the trace to the convergence of the finite-dimensional distributions. Electron. J. Probab. 23, paper no. 95 (2018).
  * [36] C. Landim, D. Marcondes, I. Seo: Metastable behavior of weakly mixing Markov chains: the case of reversible, critical zero-range processes. arXiv:2006.04214 (2020).
  * [37] C. Landim, D. Marcondes, I. Seo: A resolvent approach to metastability. arXiv: 2102.00998 (2021).
  * [38] C. Landim, M. Mariani, I. Seo:. A Dirichlet and a Thomson principle for non-selfadjoint elliptic operators, Metastability in non-reversible diffusion processes. Arch. Rational Mech. Anal. 231, 887–938 (2019) (2017)
  * [39] C. Landim, R. Misturini, K. Tsunoda: Metastability of reversible random walks in potential field. J. Stat. Phys. 160, 1449–1482 (2015)
  * [40] C. Landim, I. Seo: Metastability of non-reversible mean-field Potts model with three spins. J. Stat. Phys. 165, 693–726 (2016)
  * [41] C. Landim, I. Seo: Metastability of non-reversible random walks in a potential field, the Eyring-Kramers transition rate formula. Comm. Pure. Appl. Math. 71, 203–266 (2018)
  * [42] C. Landim, I. Seo: Metastability of one-dimensional, non-reversible diffusions with periodic boundary conditions. Ann. Inst. H. Poincaré, Probab. Statist. 55, 1850–1889 (2019).
  * [43] C. Landim, T. Xu; Metastability of finite state Markov chains: a recursive procedure to identify slow variables for model reduction. ALEA Lat. Am. J. Probab. Math. Stat. 13, 725-751 (2016)
  * [44] J. Lee: Energy Landscape and Metastability of Curie–Weiss–Potts Model J. Stat. Phys. 187, (2022)
  * [45] J. Lee, I. Seo: Non-reversible metastable diffusions with Gibbs invariant measure I: Eyring–Kramers formula. Probab. Theory Relat. Fields 182, 849–903 (2022).
  * [46] J. Lee, I. Seo: Non-reversible metastable diffusions with Gibbs invariant measure II: Markov chain convergence. arXiv:2008.08295 (2022).
  * [47] F. R. Nardi, A. Zocca: Tunneling behavior of Ising and Potts models in the low-temperature regime. Stochastic Process. Appl. 129, 4556–4575 (2019).
  * [48] E. J. Neves, R. H. Schonmann: Critical droplets and metastability for a Glauber dynamics at very low temperatures. Comm. Math. Phys. 137, 209–230 (1991).
  * [49] E. J. Neves, R. H. Schonmann: Behavior of droplets for a class of Glauber dynamics at very low temperature. Probab. Theory Related Fields 91, 331–354 (1992).
  * [50] C. Oh, F. Rezakhanlou: Metastability of zero range processes via Poisson equations. preprint available at math.berkeley.edu (2019).
  * [51] E. Olivieri and M. E. Vares. Large deviations and metastability. Encyclopedia of Mathematics and its Applications, vol. 100. Cambridge University Press, Cambridge, 2005.
  * [52] F. Rezakhanlou, I. Seo: Scaling limit of small random perturbation of dynamical systems arXiv:1812.02069 (2018).
  * [53] I. Seo: Condensation of non-reversible zero-range processes, Commun. Math. Phys. 366, 781–839 (2019)
  * [54] M. Sugiura: Metastable behaviors of diffusion processes with small parameter. J. Math. Soc. Japan 47, 755–788 (1995)
  * [55] S.R.S. Varadhan: Large Deviations and Applications. CBMS-NSF Regional Conference Series in Applied Mathematics, 46. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1984. ISBN: 0-89871-189-4


# Universal Scaling Laws for a Generic Swimmer Model

Bruno Ventéjou<EMAIL_ADDRESS>Univ. Grenoble Alpes,
CNRS, LIPhy, 38000 Grenoble, France  Thibaut Métivet<EMAIL_ADDRESS>Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France
Aurélie Dupont  Univ. Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France
Philippe Peyla<EMAIL_ADDRESS>Univ. Grenoble Alpes,
CNRS, LIPhy, 38000 Grenoble, France

###### Abstract

We have developed a minimal model of a swimmer without body deformation based
on force and torque dipoles which allows accurate 3D Navier-Stokes
calculations. Our model can reproduce swimmer propulsion for a large range of
Reynolds numbers, and generate wake vortices in the inertial regime,
reminiscent of the flow generated by the flapping tails of real fish. We
performed a numerical exploration of the model from low to high Reynolds
numbers and obtained universal laws using scaling arguments. We collected data
from a wide variety of micro-organisms, thereby extending the experimental
data presented in (M. Gazzola et al., Nature Physics 10, 758, 2014). Our
theoretical scaling laws compare very well with experimental data across the
different regimes, from Stokes to turbulent flows. We believe that this model,
due to its relatively simple design, will be very useful for obtaining
numerical simulations of collective effects within fish schools composed of
hundreds of individuals.

Introduction The wide variety of means employed by living creatures to move in
aquatic environments is fascinating Childress (1981). Motion generally
involves a complex interplay between the deformation of the body and the
surrounding fluid. From the smallest organisms, like bacteria, to colossal
blue whales Berg (2004); S.Powar  _et al._ (2022); Garcia  _et al._ (2011);
Smits (2019); Gray (1936); Wolfgang  _et al._ (1999); Dabiri  _et al._ (2006);
Müller  _et al._ (2001), the differences in length scale $L$, velocity $v$ and
mode of locomotion are so vast that the elaboration of a universal model to
describe swimming across these scales might seem impossible.

The importance of inertia with respect to viscous dissipation is quantified by
the Reynolds number, which expresses the ratio of stress due to inertia to
stress due to viscosity: $Re=\rho vL/\eta$, where $\rho$ and $\eta$ represent
fluid density and viscosity, respectively. The Reynolds numbers associated
with aquatic living species span several decades, typically ranging from
$Re^{\rm fish}\sim 10^{3}-10^{6}$ for fish to $Re^{\rm micro-org}\lesssim
10^{-3}$ for micro-organisms like spermatozoa. Consequently, the drag force
exerted by the fluid on the swimmer is largely dependent on the species
considered, originating from either fully viscous dissipation at the small
scale, or turbulent inertia at the large scale.

As a result, each swimming model Taylor (1951); Spagnolie and Lauga (2010);
Lauga (2020); Lighthill (1960); Liu and Kawachi (1999); Kern and Koumoutsakos
(2006) tends to offer a tailored approach specific to the flow regime
considered and the corresponding body deformation. Most models focus on
periodic body deformation, coupled with the surrounding fluid, and resolve the
full swimming cycle. This provides specific approaches that address swimming
at small scales, where viscous forces dominate (e.g. for micro-organisms Lauga
and Powers (2009)), differently from the macroscopic strategies of large fish
or mammals, which can leverage the inertia of the surrounding fluid to break
time reversibility Purcell (1977). The highly diverse physical origins of
particular swimming patterns represent an obstacle to the exploration of a
more comprehensive and universal viewpoint.

In this letter, we propose a different approach. Deformation kinematics, such
as undulations, oscillations and pulsations, are ignored, and locomotion is
described using force and torque dipoles applied by a solid body of finite
size $L$ on a fluid. While a similar description has already been used for
micro-swimmers at low $Re$ Hernandez  _et al._ (2005); Mehandia and Nott
(2008); Jibuti  _et al._ (2014), to the best of our knowledge, it has never
yet been employed for high $Re$, when inertia starts to dominate viscous
forces. In this work, we study the motion described by our model over $8$
decades of Reynolds numbers ($10^{-5}\lesssim Re\lesssim 10^{4}$),
theoretically and numerically, comparing our results with experimental data.
Although the direct effect on swimming velocity of the specific deformation of
the swimmer’s body is acknowledged, particularly as it can reduce the drag
force Gray (1936); Li  _et al._ (2021), we would like to emphasize that we are
not attempting to provide a detailed and precise analysis of a particular mode
of locomotion, but rather a more universal description in terms of the forces
applied to the fluid. While our swimmer model is minimal, the motion of the
surrounding fluid is accurately captured using the full numerical resolution
of the 3D Navier-Stokes equation, and our approach encompasses the different
swimming regimes of a wide variety of aquatic species. Our model can also
remarkably reproduce the characteristic wake vortices observed behind fish due
to the flapping of their tails Drucker and Lauder (2002).

Our approach furthermore exhibits universal scaling laws which link the
swimming Reynolds number $Re$ to a new dimensionless group, the thrust number
defined below. We identify three different regimes: the Stokes regime
($Re<1$), a laminar regime ($1<Re<10^{3}$) and a turbulent regime
($Re>10^{3}-10^{4}$), and calculate the theoretical exponents of the scaling
laws in the three regimes using simple scaling analyses (independently of the
space dimension). These compare very well with the numerical simulations
produced by our generic swimmer model. We also validate our results with
experimental data presented in Gazzola  _et al._ (2014) for the laminar and
turbulent regimes, and further extend this validation with data collected on
micro-swimmers for the Stokes regime.

Figure 1: The swimmer is an ellipsoidal rigid body of length $L$ and width
$L/a_{r}$, with $a_{r}$ the aspect ratio. a) Time-dependent dipoles of forces
$\\{-\bm{{F}},\bm{{F}}\\}$ and torques $\\{-\bm{{T}},\bm{{T}}\\}$ are applied
at the center-of-mass of the body $\bm{{X}}_{\mathcal{B}}$ and the “phantom
tail” $\bm{{X}}_{t}$, with $\bm{F}(t)=(\pi/2)\bm{F}_{0}\lvert\cos(\omega
t)\rvert$ and $\bm{T}(t)=\bm{T}_{0}\cos(\omega t)$. The position of
$\bm{{X}}_{t}$ can be controlled to steer the swimmer, but is kept such that
$\|\bm{{X}}_{\mathcal{B}}-\bm{{X}}_{t}\|=L/2+L/a_{r}$. b) Static model
obtained by averaging the time dependent model. Figure 2: $Re$ as a function
of $Th$ from 3D numerical simulations (crosses), obtained by solving the full
Navier-Stokes equations with our swimmer model. We clearly obtain three
regimes: $Re\sim Th^{1.0}$ for $Re\lesssim 20$; $Re\sim Th^{0.66}$ for
$20\lesssim Re\lesssim 1000$ and $Re\sim Th^{0.51}$ for $Re\gtrsim 1000$. The
three lines correspond to fitting curves and give the numerical scaling
exponents. Note that the crossovers between the different regimes depend on
the geometry of the swimmer; this curve corresponds to $a_{r}=4$.

## Swimmer model.

The model uses a time-dependent force dipole combined with a torque dipole
(Fig. 1(a)), both attached to a rigid body $\mathcal{B}$ of ellipsoidal shape.
An autonomous swimming body creates its own motion, therefore the total sum of
forces and torques must cancel out, due to the third law of Newton of the
fluid-body system. Feynman  _et al._ (2006). The model developed by Filella
_et al._ (2018) presents some conceptual similarities. It represents each fish
as a point-like active particle bearing a dipole in a potential 2D inviscid
fluid, which allows consideration of the hydrodynamic interactions between
fish in the far-field limit. However, since our aim is to explore a wide
spectrum of Reynolds numbers (from viscous to inertial regimes), we solved the
full incompressible Navier-Stokes equations in 3D and 2D (see SM ven (a)).

The swimmer exerts on the fluid a force dipole $\\{\bm{-F},\bm{F}\\}$ like
that generally used for a micro-swimmer at a low $Re$ Hernandez  _et al._
(2005); Mehandia and Nott (2008); Jibuti  _et al._ (2014) (see Fig. 1). We
used a pusher-like model that reproduces the force distribution of a fish at
high Reynolds numbers. This approach can easily be extended to a more detailed
model by using more complex force distributions (e.g. Mehandia and Nott
(2008)). As shown in Fig. 1(a), the force dipole is composed of one force
applied in the fluid at the rear of the body $\bm{{X}}_{t}$, mimicking a
swimming organ, and an opposite force exerted inside the body at the center-
of-mass $\bm{{X}}_{\mathcal{B}}$. The force is time-dependent with pulsation
$\omega$ and pusher-like: $\bm{F}(t)=(\pi/2)\bm{F_{0}}\lvert\cos\,\omega
t\rvert$ with $\mathbf{F}_{0}\cdot(\bm{{X}}_{\mathcal{B}}-\bm{{X}}_{t})>0$.
The absolute value in the expression of $\bm{F}(t)$ enforces the pusher nature
of the swimmer. We also consider a torque dipole:
$\bm{T}(t)=\bm{T_{0}}\cos\,\omega t$, collocated with the force dipole. The
torque at the back represents the stroke of the swimming organ and causes the
vortex street Drucker and Lauder (2002) in the fish’s wake at high $Re$. An
opposite torque is applied in the body (Fig. 1(a)), and represents the
counter-reaction of the rest of the body. For practical reasons in the scaling
analyses below, we also introduce force and torque densities $f_{0}\equiv
F_{0}/L^{3}$ and $\tau_{0}\equiv T_{0}/L^{3}$ respectively. Averaging these
dipoles over one period of time ($2\pi/\omega$) results in a simple static
force dipole $\\{\bm{-F_{0}},\bm{F_{0}}\\}$ (Fig. 1(b)), while the average
torque cancels out. This time averaging over one beating period is very
similar to models of pushers and pullers beating at low $Re$ Hernandez  _et
al._ (2005); Mehandia and Nott (2008); Jibuti  _et al._ (2014).

Scaling laws. The hydrodynamic nature of our model allows for simple scaling
arguments, inspired by Gazzola  _et al._ (2014) for inertial flows but
translated to a more generic framework and extended to the non-inertial Stokes
regime. In the following, we present $3D$ arguments, but they remain valid in
$2D$ (see SM ven (a)). We also consider that all the lengths scale as $L$.

The body of the swimmer is submitted to different dominant drag forces
depending on the Reynolds number. To describe this effect across all swimming
regimes, we introduce the thrust number $Th$ as the ratio between the applied
force density $f_{0}$ multiplied by inertial forces $\rho\lvert
D\mathbf{v}/Dt\rvert\sim\rho v^{2}L^{-1}$, and the square of viscous forces
$\lvert\eta\Delta\mathbf{v}\rvert^{2}\sim(\eta vL^{-2})^{2}$, which gives

$Th\equiv\frac{\rho f_{0}L^{3}}{\eta^{2}}.$ (1)

The thrust number appears naturally at all scales of Reynolds numbers, as
shown below. It contains the force term at the origin of the motion, which is
characterized by the Reynolds number. It therefore provides a convenient
method for evaluating velocity as a function of force. Let us consider classic
scaling arguments:

  * •

At high Reynolds numbers, the boundary layer around the body is turbulent and
pressure drag dominates. The corresponding force scales as $f_{0}L^{3}\sim\rho
v^{2}L^{2}$. Since $v\sim\eta Re/(\rho L)$, we obtain $Re\sim(\rho
f_{0}L^{3}/\eta^{2})^{1/2}=Th^{1/2}$.

  * •

For small but finite Reynolds numbers, the regime is laminar and the viscous
force in the boundary layer dominates: $f_{0}L^{3}\sim(\eta v/\delta)L^{2}$
where $\delta$ is the thickness of the boundary layer, which obeys the Blasius
law Landau and Lifshitz (2013) $\delta\sim L\,Re^{-1/2}$. This finally leads
to $Re\sim Th^{2/3}$.

  * •

At low Reynolds numbers, the Stokes drag force dominates and the force applied
on the body compensates the drag: $f_{0}L^{3}\sim\eta vL$. This gives
$Re\sim\rho f_{0}L^{3}/\eta^{2}=Th$.

From the Stokes to the turbulent regime, we observe that the exponent $\alpha$
of the scaling $Re\sim Th^{\alpha}$ is always below one and decreases. It
suggests a diminishing swimming performance as the Reynolds number of the
swimmer increases. To confirm these three successive regimes, we present below
numerical simulations with our swimmer model, exploring a large range of
values for the thrust and Reynolds numbers.

## Numerical simulations.

We performed direct numerical simulations of the incompressible Navier-Stokes
equations in the presence of our model swimmer, a rigid ellipsoid body with
the force and torque dipoles attached. The corresponding fluid momentum
balance equation writes:

$\displaystyle\rho\frac{D\bm{v}}{Dt}-\nabla\cdot\left(2\eta\bm{{E}}(\bm{{v}})\right)+\nabla
p=\bm{{\mathcal{F}}}+\bm{{\mathcal{T}}}$ (2)

where $\bm{{v}}$ and $p$ are respectively the velocity and pressure fields,
$D/Dt$ denotes the material derivative
$D\bm{{v}}/Dt\equiv\partial\bm{{v}}/\partial t+(\bm{v\cdot\nabla})\bm{v}$,
$\bm{{E}}(\bm{{v}})\equiv(\bm{\nabla v}+\bm{\nabla v}^{t})/2$ is the strain-
rate tensor, $\rho$ and $\eta$ denote the density and viscosity fields, and
$\bm{{\mathcal{F}}}(t)$ and $\bm{{\mathcal{T}}}(t)$ respectively represent the
– time-varying – force and torque dipoles attached to the swimmer, as
introduced above. The rigid body $\mathcal{B}$ of the swimmer is accounted for
with a fictitious domain penalty method inspired by (Janela  _et al._ , 2005);
in practice, this can be implemented simply with a spatially-variable
viscosity (Tanaka and Araki, 2000):
$\eta=\eta_{f}+(\eta_{b}-\eta_{f})\,H_{\mathcal{B}}$ where $H_{\mathcal{B}}$
is the indicator – or Heaviside – function of $\mathcal{B}$. In practice, a
viscosity ratio $\eta_{b}/\eta_{f}=10^{3}-10^{6}$ is applied between the fluid
and the swimmer body to ensure that the rigid motion constraint
$\bm{{E}}(\bm{{v}})=0$ is satisfied within $\mathcal{B}$. Density $\rho$ is
defined as constant inside and outside $\mathcal{B}$, thus making the swimmer
neutrally buoyant. Note that this fictitious domain penalty method allows the
swimmer to be treated directly as part of the Navier-Stokes equations through
the viscosity field, thereby avoiding the need to deal with moving boundary
conditions and potential remeshing issues at the body interface in the
discrete setting. The incrompressible Navier-Stokes equations are solved
numerically using an implicit $\mathcal{P}2-\mathcal{P}1$ finite element
method (Metivet  _et al._ , 2018) implemented in the parallel FEEL++ library
Prud’homme  _et al._ (2012). The position and orientation of the swimmer are
updated at each time-step using a first order Euler scheme with the
translational and rotational velocities computed from the fluid velocity field
in $\mathcal{B}$. A comprehensive derivation of the numerical model and
technical details are provided in SM ven (a).

This numerical framework is used to explore a large range of Reynolds numbers
($10^{-5}<Re<10^{4}$) and Thrust numbers ($10^{-3}<Th<10^{6})$, in order to
evaluate the dependence of $Re$ as a function of $Th$ while varying each of
the model’s different parameters ($L$, $\eta_{f}$, $f_{0}$, $\omega$ and
$\tau_{0}$) separately (see SM ven (a)). As shown in Fig. 2, the numerical
simulations are in perfect agreement with the scaling laws presented above,
displaying the Stokes regime in the range $10^{-5}\lesssim Re\lesssim 10^{2}$,
the laminar regime for $10^{2}\lesssim Re\lesssim 10^{3}$ and the turbulent
regime for $10^{3}\lesssim Re$,

Figure 3: 2D Vorticity field for different $Re$. The simulations were
performed with $\omega=2\pi$, $L=32$ and $\|\bm{{F_{0}}}\|=\|\bm{{T_{0}}}\|$.
(a): $\lVert\bm{T}_{0}\rVert=4000$ and $Re=960$. (b):
$\lVert\bm{T}_{0}\rVert=600$ and $Re=300$. (c): $\lVert\bm{T}_{0}\rVert=100$
and $Re=50$. (d): $\lVert\bm{T}_{0}\rVert=0.25$ and $Re=0.1$.

with fitting exponents that match the predictions. Note that the $Re$ ranges
of each regime depend on the aspect ratio of the swimmer $a_{r}$, which is
kept constant. We also found that $\omega$ and $\tau_{0}$ do not play any role
in the $Re(Th)$ dependency, which confirms that all the important parameters
are embedded in the $Th$ number.

Wake and vortices. Although torque plays no role in the scaling of $Re$ as a
function of $Th$, it is essential to reproduce the wake at the rear of the
swimmer in the inertial regime. The torque dipole can be used to account for
flagellum, body undulation or tail beating to generate a reverse von Karman
vortex street, as observed in the wake of a fish at high $Re$ number Kern and
Koumoutsakos (2006); Ko  _et al._ (2023).

Figures 3(a,b) illustrate the typical wakes obtained with
$(T_{0},F_{0})=(4000,4000),(600,600)$ in 2D simulations.

Figure 4: Reynolds number $Re$ as a function of the swimming number $Sw$.
Crosses are new experimental data available in the database ven (b). Dots are
experimental data collected by Gazzola  _et al._ (2014). The three lines
correspond to fitted curves and give the numerical scaling exponents of the
three different regimes: $Re\sim Sw^{0.98}$ for $Re<10$; $Re\sim Sw^{1.33}$
for $10<Re<10^{4}$; and $Re\sim Sw^{1.0}$ for $10^{4}<Re$.

The vortices created at the same frequency are spaced further apart as the
Reynolds number (i.e. velocity) increases from $Re=100$ to $Re=960$. The
length over which they dissipate becomes shorter towards the viscous regimes
(Fig. 3(c)), vanishing completely at low $Re$ (Fig. 3(d)). Indeed, no vortices
are present behind micro-organisms Drescher  _et al._ (2010).

Comparison with experimental data. To compare our scaling results with
experimental data, forces must be expressed in terms of observable data (Fig.
4), such as undulation or tail beating frequency. Although force $f_{0}$,
torque $\tau_{0}$ and pulsation $\omega$ are independent quantities in the
model, physical constraints exist between these quantities in living
organisms. We made the reasonable assumption that the size of the swimming
organ scales with the size of the body $L$ Gazzola  _et al._ (2014). Force and
torque generated by the swimming organ are such that $\tau_{0}\sim f_{0}L$.

In inertial regimes, i.e. excluding the Stokes regime that we address
separately, the instantaneous force creates a transient acceleration of the
fluid, which scales with $L\omega^{2}$, i.e. $f_{0}\sim\rho L\omega^{2}$.
Introducing the observation-based swimming number $Sw=\rho\omega L^{2}/\eta$
Gazzola  _et al._ (2014), the theoretical force drive can be related to
experimentally measurable data as $Th\sim Sw^{2}$. The scaling laws previously
derived from our model can thus be reformulated in terms of $Sw$: in the
laminar regime $Re\sim Th^{2/3}\sim Sw^{4/3}$, while in the turbulent regime
$Re\sim Th^{1/2}\sim Sw$, in accordance with the results of Gazzola  _et al._
(2014).

Figure 5: Strouhal number $St=A\omega/v$ as a function of the Reynolds number
$Re$. Dots correspond to the data collected in Gazzola  _et al._ (2014), and
crosses are new experimental data. The three lines correspond to guides for
the eyes with the three different regimes: $\tfrac{St}{2\pi}=1$ for $Re<10$;
$St\sim Re^{-0.25}$ for $10<Re<10^{4}$; and $\tfrac{St}{2\pi}\approx 0.3$ for
$10^{4}<Re$.

In the Stokes regime, the force exerted by the swimming organ is balanced by
viscous drag, so that $f_{0}L^{3}\sim\eta Lv\sim\eta L^{2}\omega$.
Reintroducing again the swimming number $Sw$, we obtain $Th\sim Sw$, leading
to $Re\sim Sw$. Note that this is a natural consequence of the absence of
inertia: each stroke creates a net displacement that scales with $\sim L$,
inducing a swimming velocity $v\sim L\omega$.

Figure 4 shows the experimental data collected from micrometer- to meter-size
aquatic organisms along with the corresponding fitted scaling laws. In
addition to the results from Gazzola  _et al._ (2014), an excellent agreement
between experimental data and hydrodynamic scaling laws is also obtained in
the Stokes regime.

Note that $v\sim L\omega$ in both the Stokes and turbulent regimes, so that
the corresponding Strouhal number $St=Sw/Re$ is constant, as illustrated in
Figure 5. The transition between $St/2\pi\approx 0.3$ Gazzola  _et al._ (2014)
and $St/2\pi\approx 1$ occurs in the laminar regime where $St\sim Re^{-1/4}$.

Conclusion. By avoiding direct fluid-structure coupling, our generic swimmer
model provides an efficient model for hydrodynamic propulsion, while retaining
the salient features of swimming organisms across several decades of Reynolds
numbers. High numerical stability and efficiency ensure fully tractable 2D and
3D simulations, thereby paving the way to large scale simulations with
hundreds of agents. It also proposes a methodology for progressive refinement
of the hydrodynamic field, by retaining higher moments of forces and torques,
resulting in more complex propulsion models. This approach broadens our
understanding of the swimming of aquatic organisms by revealing the universal
relationship between the velocity of a swimmer and the force exerted by its
swimming organ. The sub-linear dependence demonstrated between $Re$ and $Th$
suggests diminishing swimming performance as the swimmer’s Reynolds number
increases. The scaling laws obtained also match the experimental data obtained
from thousands of aquatic animals, ranging from large mammalians to micro-
organisms. Our results shed new light on the general mechanisms underlying
swimming and provide an efficient and robust numerical framework to
investigate the collective behavior of swimmers in complex environments.

Acknowledgements. This project received financial support from the French
National Research Agency (ANR-21-CE45-0005, FISHSIF project).

## References

  * Childress (1981) S. Childress,  _Mechanics of Swimming and Flying_ (Cambridge University Press, 1981).
  * Berg (2004) H. Berg,  _E.coli in motion_ (Springer, 2004).
  * S.Powar  _et al._ (2022) S.Powar, F. Parast, A. Nandagiri, A. Gaikwad, D. Potter, M. O’Bryan, R. Prabhakar, J. Soria, and R. Nosrati, Small Methods 6, e2101089 (2022).
  * Garcia  _et al._ (2011) M. Garcia, S. Berti, P. Peyla, and S. Rafaï, Phys. Rev. E 83, 035301 (2011).
  * Smits (2019) A. Smits, J. Fluid. Mech., Perspectives 874, 1 (2019).
  * Gray (1936) J. Gray, J. Exp. Biol. 13, 192 (1936).
  * Wolfgang  _et al._ (1999) M. Wolfgang, J. Anderson, M. Grosenbaugh, D. Yue, and M. Triantafyllou, J. Exp. Biol. 202, 4841 (1999).
  * Dabiri  _et al._ (2006) J. O. Dabiri, S. P. Colin, and J. H. Costello, J. Exp. Biol. 209, 17 (2006).
  * Müller  _et al._ (2001) U. Müller, J. Smit, E. Stamhuis, and J. Videler, J. Exp. Biol. 204, 2751 (2001).
  * Taylor (1951) G. I. Taylor, Proc. R. Soc. Lond. A 209, 447 (1951).
  * Spagnolie and Lauga (2010) S. Spagnolie and E. Lauga, Phys. of Fluids 22, 031901 (2010).
  * Lauga (2020) E. Lauga, Phys. Rev. Fluids 5, 123101 (2020).
  * Lighthill (1960) M. J. Lighthill, J. Fluid Mech. 9, 305 (1960).
  * Liu and Kawachi (1999) H. Liu and K. Kawachi, J. Comp. Phys. 155, 223 (1999).
  * Kern and Koumoutsakos (2006) S. Kern and P. Koumoutsakos, J. Exp. Biol. 209, 4841 (2006).
  * Lauga and Powers (2009) E. Lauga and T. R. Powers, Reports on Progess in Physics 72, 096601 (2009).
  * Purcell (1977) E. Purcell, American Journal of Physics 3–11, 1 (1977).
  * Hernandez  _et al._ (2005) J. Hernandez, C. Stoltz, and M. Graham, Phys. Rev. Lett. 95, 204501 (2005).
  * Mehandia and Nott (2008) V. Mehandia and P. Nott, J. Fluid Mech. 595, 239 (2008).
  * Jibuti  _et al._ (2014) L. Jibuti, L. Qi, C. Misbah, W. Zimmermann, S. Rafaï, and P. Peyla, Phys. Rev. E 90, 063019 (2014).
  * Li  _et al._ (2021) G. Li, H. Liu, U. Müller, C. Voesenek, and J. van Leeuwen, Proc. Royal Soc. 288, 1 (2021).
  * Drucker and Lauder (2002) E. G. Drucker and G. V. Lauder, Integrative and Comparative Biology 42, 243 (2002), https://academic.oup.com/icb/article-pdf/42/2/243/1841427/i1540-7063-042-02-0243.pdf .
  * Gazzola  _et al._ (2014) M. Gazzola, M. Argentina, and L. Mahadevan, Nature Physics 10, 758 (2014).
  * Feynman  _et al._ (2006) R. Feynman, R. Leighton, and M. Sands,  _The Feynman Lectures on Physics_ , Vol. 1 (Addison-Wesley, 2006).
  * Filella  _et al._ (2018) A. Filella, F. Nadal, C. Sire, E. Kanso, and C. Eloy, Phys. Rev. Lett. 120, 198101 (2018).
  * ven (a) “See supplementary material at [to-be-inserted-by-publisher],” (a).
  * Landau and Lifshitz (2013) L. D. Landau and E. M. Lifshitz,  _Fluid mechanics: course of theoretical physics_ , Vol. 6 (Elsevier, 2013).
  * Janela  _et al._ (2005) J. Janela, A. Lefebvre, and B. Maury, in  _ESAIM: Proceedings_ , Vol. 14 (EDP Sciences, 2005) pp. 115–123.
  * Tanaka and Araki (2000) H. Tanaka and T. Araki, Phys. Rev. Lett. 85, 1338 (2000).
  * Metivet  _et al._ (2018) T. Metivet, V. Chabannes, M. Ismail, and C. Prud’homme, Mathematics 6, 203 (2018).
  * Prud’homme  _et al._ (2012) C. Prud’homme, V. Chabannes, V. Doyeux, M. Ismail, A. Samake, and G. Pena, ESAIM: Proc. 38, 429 (2012).
  * Ko  _et al._ (2023) H. Ko, G. Lauder, and R. Nagpal, J. R. Soc. Interface 20, 1 (2023).
  * ven (b) “Doi will be added after the peer review,” (b).
  * Drescher  _et al._ (2010) K. Drescher, R. E. Goldstein, N. Michel, M. Polin, and I. Tuval, Phys. Rev. Lett. 105, 168101 (2010).