# Decomposition of class II graphs into two class I graphs††thanks: The work
of the second author has been supported by the National Science Foundation
through Grant DMS-2246292. The work of the third author has been supported by
the Simons Foundation through the Grant No. 839830. The work of the forth
author has been supported by the National Science Foundation through Grant
CCF-2008422.

Yan Cao  School of Mathematical Sciences, Dalian University of Technology,
Dalian, Liaoning, China, 116024 Guangming Jing Department of Mathematics, West
Virginia University, Morgantown, WV, USA, 26505  Rong Luo Department of
Mathematics, West Virginia University, Morgantown, WV, USA, 26505  Vahan
Mkrtchyan Corresponding Author, Computer Science Department, Boston College,
Chestnut Hill, MA, USA, 02467  Cun-Quan Zhang Department of Mathematics, West
Virginia University, Morgantown, WV, USA, 26505  Yue Zhao Department of
Mathematics, University of Central Florida, Orlando, FL, USA, 32816

###### Abstract

Mkrtchyan and Steffen [J. Graph Theory, 70 (4), 473–482, 2012] showd that
every class II simple graph can be decomposed into a maximum $\Delta$-edge-
colorable subgraph and a matching. They further conjectured that every graph
$G$ with chromatic index $\Delta(G)+k$ ($k\geq 1$) can be decomposed into a
maximum $\Delta(G)$-edge-colorable subgraph (not necessarily class I) and a
$k$-edge-colorable subgraph. In this paper, we first generalize their result
to multigraphs and show that every multigraph $G$ with multiplicity $\mu$ can
be decomposed into a maximum $\Delta(G)$-edge-colorable subgraph and a
subgraph with maximum degree at most $\mu$. Then we prove that every graph $G$
with chromatic index $\Delta(G)+k$ can be decomposed into two class I
subgraphs $H_{1}$ and $H_{2}$ such that $\Delta(H_{1})=\Delta(G)$ and
$\Delta(H_{2})=k$, which is a variation of their conjecture.

Keywords: edge-coloring, chromatic index, partition, $k$-edge-colorable
subgraph, class I graph

##  1 Introduction

Graphs considered in this paper are finite, undirected and may contain
multiple edges, but no loops. Let $V(G)$ and $E(G)$ denote the sets of
vertices and edges of a graph $G$, respectively. For a vertex $x\in V$, let
$N(x)=\\{v~{}|~{}xv\in E(G)\\}$ and
$E(x)=\\{e~{}|~{}\mbox{$e$~{}is~{}incident~{}with~{}}x\\}$. The degree of
$x\in V(G)$ is denoted by $d(x)=|E(x)|$. The minimum and maximum degrees of
vertices in $G$ are denoted by $\delta(G)$ and $\Delta(G)$, respectively. A
graph $G$ is regular if $\delta(G)=\Delta(G)$. For two vertices $x$ and $y$,
let $E(x,y)$ denote the set of edges between $x$ and $y$ and let
$\mu_{G}(x,y)=|E(x,y)|$. $\mu(G)=\max\\{\mu_{G}(x,y)|~{}x,y\in V(G)\\}$ is
called the multiplicity of $G$. If there is no confusion from the context, we
denote $\delta(G)$, $\Delta(G)$, $\mu(G)$, $\mu_{G}(x,y)$ by $\delta$,
$\Delta$, $\mu$, and $\mu(x,y)$ respectively.

An edge coloring of a graph is a function assigning values (colors) to the
edges of the graph in such a way that any two adjacent edges receive different
colors. A graph is $k$-edge-colorable if there is an edge coloring of the
graph with colors from $C=\\{1,\dots,k\\}$. The smallest integer $k$ such that
$G$ is $k$-edge-colorable is called the chromatic index of $G$, and is denoted
by $\chi^{\prime}(G)$. Clearly $\chi^{\prime}(G)\geq\Delta(G)$.

The classical theorems of Shannon and Vizing present upper bounds on
$\chi^{\prime}(G)$ as follows.

######  Theorem 1.1 (Shannon [6]).

For any graph $G$,
$\Delta(G)\leq\chi^{\prime}(G)\leq\left\lfloor\frac{3\Delta(G)}{2}\right\rfloor$.

######  Theorem 1.2 (Vizing [8]).

For any graph $G$, $\Delta(G)\leq\chi^{\prime}(G)\leq\Delta(G)+\mu(G)$.

Let $G$ be a graph with maximum degree $\Delta$. It is said to be class I if
$\chi^{\prime}(G)=\Delta$, otherwise it is class II. There are many hard
problems related to edge coloring of graphs (see [5]). A subgraph $H$ of $G$
is maximum $\Delta$-colorable if it is $\Delta$-edge-colorable and contains as
many edges as possible. The maximum $\Delta$-edge-colorable subgraphs have
been extensively studied (see [1, 2, 3, 4]). A decomposition of a graph $G$ is
a set $D=\\{H_{1},...\ ,H_{k}\\}$ of pairwise edge-disjoint subgraphs of $G$
that cover the set of edges of $G$. Since the decision problem of classifying
graphs as class I or class II is NP-complete, and every class II graph
contains a subgraph of class I, it is natural to consider how one could
decompose a class II graph into subgraphs with certain properties, for
example, into subgraphs of class I with degree conditions.

Mkrtchyan and Steffen [4] proved the following result.

######  Theorem 1.3 (Mkrtchyan and Steffen [4]).

Let $G$ be a simple graph with $\Delta(G)=\Delta$ and
$\chi^{\prime}(G)=\Delta+1$. Then $G$ can be decomposed into a maximum
$\Delta$-edge-colorable subgraph $H_{1}$ and a subgraph $H_{2}$ with
$\chi^{\prime}(H_{2})=\Delta(H_{2})=1$.

They conjectured that Theorem 1.3 can be generalized to graphs with multiple
edges.

######  Conjecture 1.4 (Mkrtchyan and Steffen [4]).

Let $G$ be a graph with $\Delta(G)=\Delta$ and $\chi^{\prime}(G)=\Delta+k$
where $k\geq 1$. Then $G$ can be decomposed into a maximum $\Delta$-edge-
colorable subgraph $H_{1}$ and a subgraph $H_{2}$ such that
$\chi^{\prime}(H_{2})=k$.

In this paper, we first generalize Theorem 1.3 as follows.

######  Theorem 1.5.

Let $G$ be a class II graph with multiplicity $\mu$ and maximum degree
$\Delta$. Then $G$ can be decomposed into two subgraphs $H_{1}$ and $H_{2}$
such that $H_{1}$ is maximum $\Delta$-edge-colorable and
$\Delta(H_{2})\leq\mu$. Moreover, if $\mu\leq 2$ and
$\chi^{\prime}(G)=\Delta+\mu$, then $\chi^{\prime}(H_{2})=\Delta(H_{2})=\mu$.

In Theorem 1.5, if $G$ is simple and class II, then $\mu=1$ and
$\Delta(H_{2})=1=\mu$. Thus it implies Theorem 1.3.

Note that a graph has chromatic index one if and only if its maximum degree is
one, and for simple graphs, it is easy to see that a maximum $\Delta$-edge-
colorable subgraph is class I by Vizing’s adjacency lemma. However, Vizing’s
adjacency lemma only works for simple graphs and consequently, as shown in
[4], maximum $\Delta$-edge-colorable subgraphs could be class II for
multigraphs. Thus in Conjecture 1.4, $H_{1}$ or $H_{2}$ could be Class II.

Our second main result decomposes an edge colored graph into two Class I
subgraphs, which is a variation of Conjecture 1.4.

######  Theorem 1.6.

Let $G$ be a graph with $\Delta(G)=\Delta$ and $\chi^{\prime}(G)=\Delta+k$
where $k\geq 1$. Then $G$ can be decomposed into two Class I subgraphs $H_{1}$
and $H_{2}$ such that $\chi^{\prime}(H_{1})=\Delta(H_{1})=\Delta$ and
$\chi^{\prime}(H_{2})=\Delta(H_{2})=k$.

The proofs of Theorems 1.5 and 1.6 will be presented in Section 3.

##  2 Preliminaries and Lemmas

In this section, we introduce additional notations and lemmas needed in the
proofs of the theorems. Let $G$ be a graph. A $k$-vertex, $k^{+}$-vertex, or
$k^{-}$-vertex is a vertex of degree $k$, at least $k$, or at most $k$,
respectively. We denote the set of all $k$-vertices, $k^{+}$-vertices, or
$k^{-}$-vertices in $V(G)$ by $V_{k}(G)$, $V_{\geq k}(G)$, or $V_{\leq k}(G)$,
respectively. For integers $r,s,t>0$, let $T=T(r,s,t)$ be the graph consisting
of three vertices $x,y,z$ such that $\mu(x,y)=r$, $\mu(y,z)=s$, and
$\mu(x,z)=t$. The graph
$S_{d}=T(\lfloor\frac{d}{2}\rfloor,\lfloor\frac{d}{2}\rfloor,\lfloor\frac{d+1}{2}\rfloor)$
is called a Shannon graph of degree $d$. Vizing [7] proved the following
structural result for graphs achieving the upper bound in Theorem 1.1.

######  Theorem 2.1 (Vizing [7]).

Let $G$ be a graph with $\Delta(G)=\Delta\geq 4$. If
$\chi^{\prime}(G)=\lfloor\frac{3}{2}\Delta\rfloor$, then $G$ contains a
Shannon graph $S_{\Delta}$ as a subgraph.

We denote $\mathcal{C}^{k}(G)$ to be the set of all $k$-edge colorings of a
graph $G$. Let $\varphi\in\mathcal{C}^{k}(G)$ and $C=\\{1,2,\dots,k\\}$ be the
color set. For a vertex $v\in V$, denote by $\varphi(v)=\\{\varphi(e)|e\in
E(v)\\}$ the set of colors present at $v$ and
$\bar{\varphi}(v)=C\setminus\varphi(v)$ the set of colors not assigned to any
edge incident with $v$. A color $\gamma$ is said to be missing at $v$ if
$\gamma\in\bar{\varphi}(v)$. For a color $\alpha$, let $E_{\alpha}=\\{e\in
E:\varphi(e)=\alpha\\}$. For an edge set $E_{0}\subseteq E(G)$, let
$\varphi(E_{0})=\\{c\in C~{}|~{}\varphi(e)=c~{}for~{}some~{}e\in E_{0}\\}$.
Let us start with the following observation.

######  Lemma 2.2.

Let $G$ be a graph with $\Delta(G)=\Delta$ and $\chi^{\prime}(G)=\Delta+k$
where $k\geq 1$. Let $x$ be a vertex in $G$. If there is a $(\Delta+k)$-edge
coloring $\varphi$ of $G$ and a vertex $v\in V(G)$ such that
$\bar{\varphi}(x)\cap\bar{\varphi}(v)=\emptyset$, then $G$ can be decomposed
into two Class I subgraphs $H_{1}$ and $H_{2}$ such that
$\Delta(H_{1})=d_{G}(x)$ and $\Delta(H_{2})=\Delta+k-d_{G}(x)$. In particular,
if $d_{G}(x)=\Delta$, then $\Delta(H_{1})=\Delta$ and $\Delta(H_{2})=k$.

###### Proof.

Assume without loss of generality that $\varphi(x)=\\{1,2,\dots,d_{G}(x)\\}$.
Let

$H_{1}=\bigcup_{i=1}^{d_{G}(x)}E_{i}~{}~{}~{}\mbox{and}~{}~{}~{}H_{2}=\bigcup_{i=d_{G}(x)+1}^{\Delta+k}E_{i}.$

Clearly $\chi^{\prime}(H_{1})\leq d_{G}(x)$ and
$\chi^{\prime}(H_{2})\leq\Delta+k-d_{G}(x)$. Note $d_{H_{1}}(x)=d_{G}(x)$ and
$\Delta+k\leq\chi^{\prime}(H_{1})+\chi^{\prime}(H_{2})$. Thus
$\chi^{\prime}(H_{1})=\Delta(H_{1})=d_{H_{1}}(x)=d_{G}(x)$. Since
$\bar{\varphi}(x)\cap\bar{\varphi}(v)=\emptyset$, we have
$d_{H_{2}}(v)=\Delta+k-d_{G}(x)$. Therefore
$\chi^{\prime}(H_{2})=\Delta(H_{2})=d_{H_{2}}(v)=\Delta+k-d_{G}(x)$, as
desired.

Let $G$ be a graph with an edge $e\in E(x,y)$, and let $\varphi$ be an edge
coloring of $G-e$. A sequence $F=(e_{1},y_{1},\ldots,e_{p},y_{p})$ consisting
of vertices and distinct edges is called a multi-fan at $x$ with respect to
$e$ and $\varphi$ if $y_{1}=y$, $e_{1}=e$, and for each $2\leq i\leq p$, we
have $e_{i}\in E(x,y_{i})$ and $\varphi(e_{i})\in\overline{\varphi}(y_{j})$
for some $1\leq j\leq i-1$. A linear sequence from $y_{1}$ to $y_{s}$, denoted
by $L=(e_{1},y_{1},e_{2},y_{2},\dots,e_{s},y_{s})$, is a sequence consisting
of distinct vertices and distinct edges such that $e_{i}\in E(x,y_{i})$ for
$1\leq i\leq s$ and $\varphi(e_{i})\in\overline{\varphi}(y_{i-1})$ for each
$2\leq i\leq p$. Clearly for any $y_{i}\in V(F)$, the multi-fan $F$ contains a
linear sequence from $y$ to $y_{i}$, as we could just pick the shortest multi-
fan contained in $F$ ending with $y_{i}$.

A shifting from $y_{1}$ to $y_{s}$ in a linear sequence
$L=(e_{1},y_{1},\dots,e_{s},y_{s})$ is an operation that obtains a new edge
coloring of $G-e_{s}$ from $\varphi$ by recoloring the edge $e_{t}$ with the
color of $e_{t+1}$ under $\varphi$ sequentially for each $1\leq t\leq s-1$ and
uncoloring the edge $e_{s}$.

An edge $e\in E(x,y)$ is called a critical edge of $G$ if
$\chi^{\prime}(G-e)<\chi^{\prime}(G)$. Clearly if $e$ is critical, then
$\chi^{\prime}(G-e)=\chi^{\prime}(G)-1$. We first have the following result
regarding multi-fans.

######  Lemma 2.3.

Let $G$ be a graph with $\Delta(G)=\Delta$ and multiplicity $\mu$. Suppose
that $\chi^{\prime}(G)\geq\Delta+1$ and $e\in E(x,y)$ is a critical edge of
$G$. Let $F=(e_{1},y_{1},\ldots,e_{p},y_{p})$ be a maximal multi-fan at $x$
with respect to $e$ and $\varphi\in\mathcal{C}^{\chi^{\prime}(G)-1}(G-e)$.
Then we have the following:

(a) (Stiebitz et al. [5]) $|V(F)|\geq 2$ and $\sum_{z\in
V(F)}(d_{G}(z)+\mu_{F}(x,z)-(\chi^{\prime}(G)-1))=2$, where $\mu_{F}(x,z)$ is
the number of edges between $x$ and $z$ in $F$.

(b) If $d_{G}(y)\leq\chi^{\prime}(G)-\mu$, then $F$ contains a vertex $z\neq
y$ with $d_{G}(z)\geq\chi^{\prime}(G)-\mu$.

###### Proof.

(a) is Theorem 2.1 in book [5] due to Stiebitz et al.

(b) Since $\mu\geq\mu_{F}(x,z)$, by (a) we have

$2=\sum_{z\in V(F)}(d_{G}(z)+\mu_{F}(x,z)-(\chi^{\prime}(G)-1))\leq\sum_{z\in
V(F)}(d_{G}(z)+\mu-(\chi^{\prime}(G)-1)).$

Thus $F$ contains either a vertex $z$ with
$d_{G}(z)+\mu-(\chi^{\prime}(G)-1)>1$, or two vertices $z_{i}$ with $i=1,2$
such that $d_{G}(z_{i})+\mu-(\chi^{\prime}(G)-1)=1$. In the former case we
have $d_{G}(z)>(\chi^{\prime}(G)-1)-\mu+1=\chi^{\prime}(G)-\mu$, so $z\neq y$
(because $d_{G}(y)\leq\chi^{\prime}(G)-\mu$). In the latter case we have
$d_{G}(z_{i})\geq(\chi^{\prime}(G)-1)-\mu+1=\chi^{\prime}(G)-\mu$ and one of
$z_{1}$ or $z_{2}$ is not $y$, as desired.

##  3 Decomposition of class II graphs

In this section, we prove Theorems 1.5 and 1.6. We first would like to point
out that the proof would be much easier if there is no restriction on the
maximum degrees of $H_{1}$ and $H_{2}$.

######  Proposition 3.1.

Every class II graph can be decomposed into two class I graphs.

###### Proof.

Let $G$ be a graph with maximum degree $\Delta$ and
$\chi^{\prime}(G)=\Delta+k$ where $k\geq 1$. Let
$\varphi\in\mathcal{C}^{\Delta+k}(G)$ such that $|E_{1}|$ is minimum, where
$E_{1}$ is the set of edges in $G$ colored by the color $1$. Let $e\in E(x,y)$
and $e\in E_{1}$.

By the minimality of $|E_{1}|$, we have
$\bar{\varphi}(x)\cap\bar{\varphi}(y)=\emptyset$. Otherwise one may recolor
the edge $e$ with a color $i\in\bar{\varphi}(x)\cap\bar{\varphi}(y)$, which
contradicts the minimality of $|E_{1}|$. By Lemma 2.2, $G$ can be decomposed
into two class I subgraphs $H_{1}$ and $H_{2}$ such that
$\Delta(H_{1})=d_{G}(x)$ and $\Delta(H_{2})=\Delta+k-d_{G}(x)$.

We are ready to prove Theorem 1.5.

Theorem 1.5 Let $G$ be a class II graph with multiplicity $\mu$ and maximum
degree $\Delta$. Then $G$ can be decomposed into two subgraphs $H_{1}$ and
$H_{2}$ such that $H_{1}$ is maximum $\Delta$-edge-colorable and
$\Delta(H_{2})\leq\mu$. Moreover, if $\mu\leq 2$ and
$\chi^{\prime}(G)=\Delta+\mu$, then $\chi^{\prime}(H_{2})=\Delta(H_{2})=\mu$.

###### Proof.

We first prove the general case. Take a maximum $\Delta$-edge-colorable
subgraph $H_{1}$ of $G$ minimizing

$t(H_{2})=\sum_{d_{H_{2}}(v)>\mu}d_{H_{2}}(v),$

where $H_{2}=G-H_{1}$. It is sufficient to show $t(H_{2})=0$. Suppose to the
contrary that $t(H_{2})\geq 1$. Let $y$ be a vertex with $d_{H_{2}}(y)>\mu$
and $e\in E(x,y)$ be an edge in $H_{2}$.

Since $H_{1}$ is maximum $\Delta$-edge-colorable, the edge $e$ is critical in
$G^{\prime}=H_{1}+e$ and $\chi^{\prime}(G^{\prime})=\Delta+1$. Let
$\varphi\in\mathcal{C}^{\Delta}(H_{1})$ and $F$ be a maximal multi-fan at $x$
with respect to $e$ and $\varphi$. Since $d_{H_{2}}(y)>\mu$ and
$d_{G}(y)\leq\Delta$, we have

$d_{G^{\prime}}(y)=d_{H_{1}}(y)+1=d_{G}(y)-d_{H_{2}}(y)+1\leq\Delta-(\mu+1)+1<\Delta-\mu+1\leq\chi^{\prime}(G^{\prime})-\mu(G^{\prime}).$

By Lemma 2.3 (b) with
$\chi^{\prime}(G^{\prime})=\Delta+1\geq\Delta(G^{\prime})+1$, there is a
vertex $z\in V(F)\setminus\\{y\\}$ such that
$d_{H_{1}}(z)\geq\chi^{\prime}(G^{\prime})-\mu(G^{\prime})\geq\Delta+1-\mu$.
This implies $d_{H_{2}}(z)\leq\mu-1$.

By the definition of a multi-fan, $F$ has a linear sequence $L$ from $y$ to
$z$ with the last edge $f\in E(x,z)$. By shifting colors along $L$, we see
that $H_{1}^{\prime}=G^{\prime}-f=H_{1}+e-f$ is $\Delta$-edge-colorable and
has the same number of edges as $H_{1}$. Thus $H_{1}^{\prime}$ is also maximum
$\Delta$-edge-colorable. Let $H_{2}^{\prime}=G-H_{1}^{\prime}$. Then
$d_{H_{2}}(w)=d_{H_{2}^{\prime}}(w)$ for each vertex $w\not=y,z$,
$d_{H_{2}^{\prime}}(y)=d_{H_{2}}(y)-1\geq\mu$ and
$d_{H_{2}^{\prime}}(z)=d_{H_{2}}(z)+1$. Since $d_{H_{2}}(z)\leq\mu-1$, we have
$d_{H_{2}^{\prime}}(z)\leq\mu$. Therefore

$t(H_{2}^{\prime})=\left\\{\begin{array}[]{ll}t(H_{2})-1&\mbox{if
$d_{H_{2}}(y)>\mu+1$, }\\\ t(H_{2})-d_{H_{2}}(y)&\mbox{ if
$d_{H_{2}}(y)=\mu+1$.}\end{array}\right.$

In either case we have $t(H_{2}^{\prime})<t(H_{2})$, which contradicts the
choice of $H_{1}$. This proves the general case.

Now we prove the “moreover” part. Assume that $\mu\leq 2$ and
$\chi^{\prime}(G)=\Delta+\mu$. By the first part, let $H_{1}$ be a maximum
$\Delta$-edge-colorable subgraph, such that
$\Delta(H_{2})=\Delta(G-H_{1})\leq\mu$ where $H_{2}=G-H_{1}$, and the number
of odd cycles in $H_{2}$ is smallest. We will show
$\chi^{\prime}(H_{2})=\Delta(H_{2})=\mu$. Note $1\leq\Delta(H_{2})\leq\mu\leq
2$.

If $\Delta(H_{2})=1$, then $\chi^{\prime}(H_{2})=1$ and thus
$\chi^{\prime}(G)=\Delta+1$, so $\mu=1$. Therefore
$\chi^{\prime}(H_{2})=\Delta(H_{2})=\mu$.

Now assume $\Delta(H_{2})=2$. Then $\mu=2$ and $H_{2}$ consists of vertex
disjoint cycles and paths. To show $\chi^{\prime}(H_{2})=2$, it is sufficient
to show that $H_{2}$ contains no odd cycles. Suppose by contradiction that
$H_{2}$ contains an odd cycle and $e\in E(x,y)$ is an edge in $E(H_{2})$ lying
in an odd cycle of $H_{2}$.

Similar to the argument in the first part, $e$ is critical in
$G^{\prime}=H_{1}+e$ and $\chi^{\prime}(G^{\prime})=\Delta+1$. Let
$\varphi\in\mathcal{C}^{\Delta}(H_{1})$ and $F$ be a maximal multi-fan at $x$
with respect to $e$ and $\varphi$. Since $d_{H_{2}}(y)=2$ and
$d_{G}(y)\leq\Delta$, we have
$d_{G^{\prime}}(y)\leq\Delta-1=\Delta-\mu+1\leq\chi^{\prime}(G^{\prime})-\mu(G^{\prime})$.
Then by Lemma 2.3 with $\chi^{\prime}(G)=\Delta+1\geq\Delta(G^{\prime})+1$,
$F$ has a vertex $z\neq y$ such that
$d_{H_{1}}(z)\geq\chi^{\prime}(G)-\mu(G^{\prime})\geq\Delta-1$. So
$d_{H_{2}}(z)\leq 1$. This implies that $z$ is an endvertex of a path or a
$0$-vertex in $H_{2}$. Again $F$ has a linear sequence $L$ from $y$ to $z$
with last edge $f\in E(x,z)$, and by shifting colors along $L$, we see that
$H_{1}^{\prime}=G^{\prime}-f=H_{1}+e-f$ is maximum $\Delta$-edge-colorable. We
have that $\Delta(H_{2}^{\prime})=2$, where
$H_{2}^{\prime}=G-H_{1}^{\prime}=H_{2}-e+f$, and the number of odd cycles in
$H^{\prime}_{2}$ is one less than in $H_{2}$. This contradicts the choice of
$H_{1}$ and $H_{2}$ and thus proves the “moreover” part.

For the proof of Theorem 1.6, we introduce additional notations. Let
$\varphi\in\mathcal{C}^{k}(G)$. For any two distinct colors $\alpha$ and
$\beta$, let $G_{\varphi}(\alpha,\beta)$ be the subgraph of $G$ induced by
$E_{\alpha}\cup E_{\beta}$. The components of $G_{\varphi}(\alpha,\beta)$ are
called $(\alpha,\beta)$-chains. Clearly, each $(\alpha,\beta)$-chain is either
a path or a cycle of edges alternately colored with $\alpha$ and $\beta$. For
each $(\alpha,\beta)$-chain $P$, let $\varphi/P$ denote the $k$-edge-coloring
of $G$ obtained from $\varphi$ by exchanging colors $\alpha$ and $\beta$ on
$P$.

For any $v\in V(G_{\varphi}(\alpha,\beta))$, let $P_{v}(\alpha,\beta,\varphi)$
denote the unique $(\alpha,\beta)$-chain containing $v$. Note that, for any
two vertices $u,\,v\in V(G_{\varphi}(\alpha,\beta))$, either
$P_{u}(\alpha,\beta,\varphi)=P_{v}(\alpha,\beta,\varphi)$ or
$P_{u}(\alpha,\beta,\varphi)$ is vertex-disjoint from
$P_{v}(\alpha,\beta,\varphi)$. This fact will be used very often without
mentioning. We are ready to prove Theorem 1.6.

Theorem 1.6 Let $G$ be a graph with $\Delta(G)=\Delta$ and
$\chi^{\prime}(G)=\Delta+k$ where $k\geq 1$. Then $G$ can be decomposed into
two Class I subgraphs $H_{1}$ and $H_{2}$ such that
$\chi^{\prime}(H_{1})=\Delta(H_{1})=\Delta$ and
$\chi^{\prime}(H_{2})=\Delta(H_{2})=k$.

###### Proof.

We prove the theorem by contradiction. Let $G$ be a counterexample minimizing
$\Delta=\Delta(G)$. Clearly, $\Delta\geq 3$.

Let $x\in V(G)$ be a $\Delta$-vertex. We have the following claim.

######  Claim 1.

For any $y\in V_{\geq\Delta-1}(G)\setminus\\{x\\}$, we have $|E(x,y)|\geq
d(y)-k+1$. Consequently $V_{\geq\Delta-1}(G)\subseteq N(x)\cup\\{x\\}$.

###### Proof.

Suppose to the contrary that $|E(x,y)|<d(y)-k+1$. Let $\varphi\in
C^{\Delta+k}(G)$ be a coloring such that
$|\bar{\varphi}(x)\cap\bar{\varphi}(y)|$ is minimum.

Define:

$C_{1}=\bar{\varphi}(x)\cap\bar{\varphi}(y)=\overline{\varphi(x)\cup\varphi(y)},$
$C_{2}=\bar{\varphi}(x)\setminus\bar{\varphi}(y)=\varphi(y)\setminus\varphi(x),$
$C_{3}=\bar{\varphi}(y)\setminus\bar{\varphi}(x)=\varphi(x)\setminus\varphi(y),$
$~{}~{}~{}\mbox{and}~{}~{}C_{4}=\varphi(x)\cap\varphi(y).$

It is easy to see that $C_{1},C_{2},C_{3},C_{4}$ are pairwise disjoint and
form a partition of the color set $C=\\{1,2,\dots,\Delta+k\\}$. By Lemma 2.2,
we have $C_{1}=\bar{\varphi}(x)\cap\bar{\varphi}(y)\not=\emptyset$.

######  Subclaim 1.1.

$P_{x}(\alpha,\eta,\varphi)=P_{y}(\alpha,\eta,\varphi)$ for any two colors
$\alpha\in C_{1}$ and $\eta\in C_{4}$.

###### Proof.

Otherwise let $\varphi^{\prime}=\varphi/P_{y}(\alpha,\eta,\varphi)$. Then
$\bar{\varphi}^{\prime}(x)\cap\bar{\varphi}^{\prime}(y)=(\bar{\varphi}(x)\cap\bar{\varphi}(y))\setminus\\{\alpha\\}$,
a contradiction to the minimality of $|\bar{\varphi}(x)\cap\bar{\varphi}(y)|$.

######  Subclaim 1.2.

$\varphi(E(x)\setminus E(x,y))\cap C_{4}\not=\emptyset$.

###### Proof.

Suppose to the contrary $\varphi(E(x)\setminus E(x,y))\cap C_{4}=\emptyset$.
Since $\varphi(E(x,y))\subseteq C_{4}$, we have $C_{4}=\varphi(E(x,y))$. This
implies that $C_{2}=\varphi(E(y)\setminus E(x,y))$ and
$C_{3}=\varphi(E(x)\setminus E(x,y))$. Thus $|C_{4}|=|E(x,y)|$,
$|C_{2}|=d(y)-|E(x,y)|$, and $|C_{3}|=d(x)-|E(x,y)|=\Delta-|E(x,y)|$.
Therefore

$\displaystyle\Delta+k=|C|$ $\displaystyle=|C_{1}|+|C_{2}|+|C_{3}|+|C_{4}|$
$\displaystyle=|C_{1}|+\Delta+d(y)-|E(x,y)|$
$\displaystyle>1+\Delta+d(y)-(d(y)-k+1)$ $\displaystyle=\Delta+k.$

This contradiction proves the subclaim.

By Subclaim 1.2, let $z_{1}\in N(x)\setminus\\{y\\}$ and $e_{1}\in E(x,z_{1})$
such that $\varphi(e_{1})=\eta\in C_{4}$.

######  Subclaim 1.3.

$\bar{\varphi}(z_{1})\cap(C_{1}\cup C_{4})=\emptyset$. Therefore
$\bar{\varphi}(z_{1})\subseteq C_{2}\cup C_{3}$.

###### Proof.

Suppose to the contrary that there exists a color
$\alpha\in\bar{\varphi}(z_{1})\cap(C_{1}\cup C_{4})$.

If $\alpha\in C_{1}$, then
$P_{x}(\alpha,\eta,\varphi)=P_{z_{1}}(\alpha,\eta,\varphi)=xz_{1}$, a
contradiction to Subclaim 1.1.

Assume $\alpha\in C_{4}$. Let $\beta\in C_{1}$. Then
$P_{z_{1}}(\beta,\alpha,\varphi)$ does not contain $x,y$ by Subclaim 1.1. Let
$\varphi^{\prime}=\varphi/P_{z_{1}}(\beta,\alpha,\varphi)$. Then
$\beta\in\bar{\varphi}^{\prime}(z_{1})$ and
$\beta\in\bar{\varphi}^{\prime}(x)\cap\bar{\varphi}^{\prime}(y)$. Note
$\bar{\varphi}^{\prime}(x)\cap\bar{\varphi}^{\prime}(y)=\bar{\varphi}(x)\cap\bar{\varphi}(y)$.
Thus we are back to the previous case.

By Subclaim 1.3, $\bar{\varphi}(z_{1})\subseteq C_{2}\cup C_{3}$. Since
$|\bar{\varphi}(z_{1})|=\Delta+k-d(z_{1})\geq k$ and
$|C_{2}|=|\bar{\varphi}(x)|-|C_{1}|=k-|C_{1}|<k$, we have
$\bar{\varphi}(z_{1})\cap C_{3}\not=\emptyset$.

Let $\gamma_{1}$ be a color in $\bar{\varphi}(z_{1})\cap C_{3}$. Then there
are $z_{2}\in N(x)$ and $e_{2}\in E(x,z_{2})$ such that
$\varphi(xz_{2})=\gamma_{1}$. Since
$\gamma_{1}\in\bar{\varphi}(z_{1})\cap\bar{\varphi}(y)$, we have
$z_{2}\not=z_{1}$ and $z_{2}\not=y$.

######  Subclaim 1.4.

$\bar{\varphi}(z_{2})\subseteq C_{2}\cup
C_{3}=(\bar{\varphi}(x)\cup\bar{\varphi}(y))\setminus(\bar{\varphi}(x)\cap\bar{\varphi}(y))$.

###### Proof.

Suppose to the contrary that there is a color $\alpha\in\bar{\varphi}(z_{2})$
with $\alpha\not\in C_{2}\cup C_{3}$.

If $\alpha\in C_{1}$, then
$P_{x}(\alpha,\gamma_{1},\varphi)=P_{z_{2}}(\alpha,\gamma_{1},\varphi)=xz_{2}$.
Thus $P_{z_{1}}(\alpha,\gamma_{1},\varphi)$ does not contain $x$. Note that
both $\alpha$ and $\gamma_{1}$ are missing at $y$. Let
$\varphi^{\prime}=\varphi/P_{z_{1}}(\alpha,\gamma_{1},\varphi)$. Then
$\alpha\in\bar{\varphi}^{\prime}(z_{1})$ and
$\alpha\in\bar{\varphi}^{\prime}(x)\cap\bar{\varphi}^{\prime}(y)$, a
contradiction to Subclaim 1.3.

Now assume $\alpha\in C_{4}$. Let $\beta$ be a color in $C_{1}$. Then
$P_{z_{2}}(\beta,\alpha,\varphi)$ does not contain $x$ or $y$ by Subclaim 1.1.
Let $\varphi^{\prime}=\varphi/P_{z_{2}}(\beta,\alpha,\varphi)$. Then
$\beta\in\bar{\varphi}^{\prime}(z_{2})$ and
$\beta\in\bar{\varphi}^{\prime}(x)\cap\bar{\varphi}^{\prime}(y)=C_{1}$, where
we have reached the case above with $\beta$ replacing $\alpha$.

Note that
$|\bar{\varphi}(z_{1})|+|\bar{\varphi}(z_{2})|=2(\Delta+k)-d(z_{1})-d(z_{2})\geq
2k$ and $|C_{2}\cup C_{3}|=|\bar{\varphi}(x)|+|\bar{\varphi}(y)|-2|C_{1}|\leq
k+(k+1)-2=2k-1$. By Subclaims 1.3 and 1.4,
$\bar{\varphi}(z_{1})\cup\bar{\varphi}(z_{2})\subseteq C_{2}\cup C_{3}$. Thus
$\bar{\varphi}(z_{1})\cap\bar{\varphi}(z_{2})\not=\emptyset$. Let $\beta$ be a
color in $\bar{\varphi}(z_{1})\cap\bar{\varphi}(z_{2})$. Then $\beta\in
C_{2}\cup C_{3}$. Let $\alpha$ be a color in $C_{1}$.

If $\beta\in C_{2}$, then $\beta,\alpha\in\bar{\varphi}(x)$ and at least one
of $P_{z_{1}}(\beta,\alpha,\varphi),P_{z_{2}}(\beta,\alpha,\varphi)$ does not
contain $y$.

If $\beta\in C_{3}$, then $\beta,\alpha\in\bar{\varphi}(y)$ and at least one
of $P_{z_{1}}(\beta,\alpha,\varphi)$ and $P_{z_{2}}(\beta,\alpha,\varphi)$
does not contain $x$.

In either case, at least one of
$P_{z_{1}}(\beta,\alpha,\varphi),P_{z_{2}}(\beta,\alpha,\varphi)$ does not
contain $x$ or $y$.

If $P_{z_{1}}(\beta,\alpha,\varphi)$ does not contain $x$ or $y$, let
$\varphi^{\prime}=\varphi/P_{z_{1}}(\beta,\alpha,\varphi)$. Then
$\alpha\in\bar{\varphi}^{\prime}(z_{1})$ and
$\alpha\in\bar{\varphi}^{\prime}(x)\cap\bar{\varphi}^{\prime}(y)$, a
contradiction to Subclaim 1.3.

If $P_{z_{2}}(\beta,\alpha,\varphi)$ does not contain $x$ or $y$, let
$\varphi^{\prime}=\varphi/P_{z_{2}}(\beta,\alpha,\varphi)$. Then
$\alpha\in\bar{\varphi}^{\prime}(z_{2})$ and
$\alpha\in\bar{\varphi}^{\prime}(x)\cap\bar{\varphi}^{\prime}(y)$, a
contradiction to Subclaim 1.4. This completes the proof of Claim 1.

By Claim 1, every $(\Delta-1)^{+}$-vertex is either $x$ or is adjacent to $x$.
The next claim shows that there are at most two $(\Delta-1)^{+}$-vertices in
$G$.

######  Claim 2.

There is at most one $(\Delta-1)^{+}$-vertex in $N(x)$.

###### Proof.

Suppose to the contrary that there exist two $(\Delta-1)^{+}$-vertices $y,z\in
N(x)$. By Claim 1,

$\Delta=d(x)\geq|E(x,y)|+|E(x,z)|\geq d(y)-k+1+d(z)-k+1\geq 2\Delta-2k.$

Thus $k\geq\frac{\Delta}{2}$. On the other hand, we know
$k\leq\frac{\Delta}{2}$ by Theorem 1.1. This implies that
$k=\frac{\Delta}{2}$, $\Delta$ is even, $d(y)=d(z)=\Delta-1$, and
$N(x)=\\{y,z\\}$. Thus by Claim 1, we have $V_{\geq\Delta-1}(G)=\\{x,y,z\\}$.
However, Theorem 2.1 implies that $G$ must contain a Shannon graph of degree
$\Delta$ as a subgraph which contains at least three $\Delta$-vertices, a
contradiction.

Let $y\in N(x)$ such that $d_{G}(y)=\max\\{d_{G}(z)|z\in N(x)\\}$. Define
$e\in E(x,y)$. Then, each vertex in $V(G)-\\{x,y\\}$ has degree at most
$\Delta-2$. Let $\varphi\in\mathcal{C}^{\Delta+k}(G)$ and $\alpha=\varphi(e)$.
Let $G^{\prime}=G-E_{\alpha}$. Then $\Delta(G^{\prime})=\Delta-1$ and
$\chi^{\prime}(G^{\prime})=\Delta+k-1$. Thus by the minimality of $\Delta$,
$G^{\prime}=G-E_{\alpha}$ can be decomposed into two class I subgraphs
$H_{1}^{\prime}$ and $H_{2}^{\prime}$ such that
$\Delta(H_{1}^{\prime})=\Delta-1$ and $\Delta(H_{2}^{\prime})=k$. Let
$H_{1}=H_{1}^{\prime}+E_{\alpha}$ and $H_{2}=H_{2}^{\prime}$. Since
$V_{\geq\Delta-1}(G)\subseteq\\{x,y\\}$, we have
$\Delta(H_{1})=\Delta(H_{1}^{\prime})+1=\Delta$. Thus $H_{1}$ and $H_{2}$ form
a desired decomposition. This completes the proof of Theorem 1.6.

##  4 Concluding remarks

Theorem 1.6 states that the maximum degree of the two subgraphs are $\Delta$
and $k$ respectively. We believe that it can be generalized as follows.

######  Conjecture 4.1.

Let $G$ be a graph with $\chi^{\prime}(G)=\Delta+k$ and $p,q$ be two positive
integers such that $p+q=\Delta+k$ with $p,q\leq\Delta$. Then $G$ can be
decomposed into two class I subgraphs $H_{1}$ and $H_{2}$ such that
$\Delta(H_{1})=p$ and $\Delta(H_{2})=q$.

Regarding maximum $\Delta$-edge-colorable subgraphs, Mkrtchyan and Steffen [4]
proved the following.

######  Theorem 4.2 (Mkrtchyan and Steffen [4]).

Let $G$ be a bridgeless cubic graph and $M$ be a perfect matching of $G$. Then
there is a maximum $3$-edge-colorable subgraph $H$ such that $M\cup
E(H)=E(G)$.

We believe that Theorem 4.2 can be generalized as follows:

######  Conjecture 4.3.

Let $G$ be a simple graph. Then for each maximum matching $M$ of $G$, there is
a maximum $\Delta$-colorable subgraph $H$ of $G$ with $M\cup E(H)=E(G)$.

######  Conjecture 4.4.

Let $G$ be an $r$-regular graph with $\chi^{\prime}(G)=\Delta+1$. Then for
each maximum matching $M$ of $G$, there is a maximum $\Delta$-colorable
subgraph $H$ of $G$ with $M\cup E(H)=E(G)$.

## References

  * [1] M. Albertson and R. Haas, Parsimonious edge coloring, Discrete Mathematics, (148):1–7, 1996.
  * [2] R. Rizzi, Approximating the maximum 3-edge-colorable subgraph problem, Discrete Mathematics, 309(12):4166–4170, 2009.
  * [3] V. Mkrtchyan, S. Petrosyan, and G. Vardanyan, On disjoint matchings in cubic graphs, Discrete Mathematics, (310):1588–1613, 2010.
  * [4] V. V. Mkrtchyan and E. Steffen, Maximum $\Delta$-edge-colorable subgraphs of class II graphs, J. Graph Theory, 70 (4), 473–482, 2012.
  * [5] M. Stiebitz, D. Scheide, B. Toft, and L. M. Favrholdt, Graph Edge Coloring, John Wiley and Sons, 2012.
  * [6] C. E. Shannon, A theorem on coloring the lines of a network, J. Math. Physics, (28): 148–151, 1949.
  * [7] V. G. Vizing, The chromatic class of a multigraph (in Russian), _Kibernetika (Kiev)_ , 3: 29-39. English translation in: _Cybernetics and System Analysis_ , 1:32-41.
  * [8] V. Vizing, On an estimate of the chromatic class of a $p$-graph, Diskret Analiz, (3):25–30, 1964.


# Full Statistics of Regularized Local Energy Density in a Freely Expanding
Kipnis-Marchioro-Presutti Gas

Eldad Bettelheim<EMAIL_ADDRESS>Racah Institute of Physics,
Hebrew University of Jerusalem, Jerusalem 91904, Israel  Baruch Meerson
<EMAIL_ADDRESS>Racah Institute of Physics, Hebrew University of
Jerusalem, Jerusalem 91904, Israel

###### Abstract

We combine the Macroscopic Fluctuation Theory and the Inverse Scattering
Method to determine the full long-time statistics of the energy density
$u(x,t)$ averaged over a given spatial interval,

$U=\frac{1}{2L}\int_{-L}^{L}dx\,u(x,t)\,,$

in a freely expanding Kipnis-Marchioro-Presutti (KMP) lattice gas on the line,
following the release at $t=0$ of a finite amount of energy at the origin. In
particular, we show that, as time $t$ goes to infinity at fixed $L$, the large
deviation function of $U$ approaches a universal form when expressed in terms
of the energy content of the interval $|x|<L$ at time $t$. A key part of the
solution is the determination of the most likely configuration of the energy
density at time $t$, conditional on $U$.

##  I Introduction and MFT equations

Stochastic lattice gases are simple, versatile and instructive models which
capture universal aspects of large deviations of different fluctuating
quantities (the density field, the integrated current, etc.) in macroscopic
systems of interacting particles out of equilibrium Spohn ; Liggett ; KL ;
Krapivskybook . The last two decades have witnessed a major progress in this
area of nonequilibrium statistical mechanics. To a large extent, this progress
has been made possible by the development of fluctuational hydrodynamics (FHD)
Spohn  and macroscopic fluctuation theory (MFT) JonaLasinioreview . Stripped
off irrelevant microscopic details, FHD and MFT are ideally suited for probing
late-time asymptotic regimes which are usually the most interesting. This is
especially true for non-stationary processes, where exact results for full
statistics – by any method – are scarce DG2009a ; DG2009b ; IMS2017 ; IMS2021
; BSM2022a ; Grabsch2022 ; Mallick2022 ; BSM2022b ; KLD2023 ; BM2024 .

Here we focus on the continuous-time Kipnis-Marchioro-Presutti (KMP) lattice
gas model KMP . It involves immobile “agents” occupying a lattice and carrying
continuous non-zero amounts of “energy”. At each random move a randomly chosen
pair of nearest neighbors randomly redistributes their combined energy among
them according to uniform distribution. The original motivation behind the KMP
model was a rigorous proof of Fourier’s law of heat diffusion when starting
from a microscopic model KMP . Since then the KMP model has been extensively
studied in the context of nonequilibrium fluctuations of transport Bertini2005
; BGL ; BodineauDerrida ; DG2009b ; Lecomte ; Tailleur ; HurtadoGarrido ; KrMe
; Pradosetal ; MS2013 ; ZarfatyM ; Peletier ; Spielberg ; Guttierez ; Frassek
; Benichou .

In a parallel development, the last 15 years have witnessed significant
interest and remarkable advancements Corwin2012 ; Spohn2015 ; QuastelSpohn2015
; Takeuchi2018 ; KK ; MKV2016 ; JKM2016 ; SMP2017 ; SmithMeerson2018 ; KLD2021
; KLD2022  in the studies of the _one-point_ statistics of the Kardar-Parisi-
Zhang (KPZ) equation KPZ , a paradigmatic model of nonconservative
nonequilibrium stochastic growth. An analogous one-point statistics in a
stochastic lattice gas would be the statistics of the gas density (or
temperature) at a specified point in space. It turns out, however, that in the
continuum description of the conservative lattice gases, provided by the FHD,
the one-point statistics is ill-defined, as the variance of the one-point
fluctuations diverges at small scales. A similar ultraviolet catastrophe of
the one-point fluctuations in nonequilibrium stochastic growth is well known
to experts Krug ; SmithMS . It occurs in a broad family of continuous
stochastic growth models when the dimension of space exceeds a model-dependent
critical dimension SmithMS . Fortunately, for the KPZ equation the critical
dimension is $2$, which allows for unhindered studies of the one-point
statistics in one dimension. For the lattice gases, however, the ultraviolet
catastrophe occurs in all physical dimensions.

One remedy against the ultraviolet catastrophe is to introduce a small-scale
cutoff: most naturally, the lattice constant. Such a regularization, however,
would imply abandoning continuous theory with many of its benefits. An
advantageous regularization alternative is to characterize local fluctuations
by the probability distribution $\mathcal{P}(U,T)$ of the density or energy
density averaged over a small but _macroscopic_ spatial interval, $|x|<L$, at
time $t=T$:

$\frac{1}{2L}\int_{-L}^{L}dx\,u(x,t=T)=U\,.$ (1)

This type of regularization was proposed in Ref. SmithMS  for the stochastic
surface growth. Here we apply it to a freely expanding KMP gas with a finite
total energy $W$. Since we are interested in the long-time limit,
$T\to\infty$, we may assume that the energy is initially released at a point,
which we choose for simplicity to be $x=0$:

$u(x,t=0)=W\delta(x)\,.$ (2)

The _expected_ value of the locally averaged energy density $U$ at the same
point at time $T$,

$\bar{U}=\frac{W\text{erf}\left(\frac{L}{\sqrt{4T}}\right)}{2L}\,,$ (3)

readily follows from the mean-field (that is, zero-noise) solution,

$\bar{u}(x,t)=\frac{W}{\sqrt{4\pi t}}\,e^{-\frac{x^{2}}{4t}}\,,$ (4)

of the diffusion equation
$\partial_{t}\bar{u}(x,t)=\partial_{x}^{2}\bar{u}(x,t)$ that governs the mean
coarse-grained energy density of the KMP gas at long times KMP ; Spohn ; KL .
Here we study fluctuations, including large deviations, of $U$ around the
expected value (3). In the long-time limit, $T\gg L^{2}$, the error function
in Eq. (3) can be expanded at small argument, and we obtain

$\bar{U}\simeq\frac{W}{\sqrt{4\pi T}}\,,$ (5)

which is independent of the regularization length $2L$.

It is convenient to re-scale $t$, $x$ and $u$ by $T$, $\sqrt{T}$ and
$W/\sqrt{T}$, respectively. The initial condition (2) becomes

$u(x,t=0)=\delta(x)\,,$ (6)

while the condition (1) on the locally averaged density at $t=T$ can be
written as

$\int_{-\ell}^{\ell}dx\,u(x,t=1)=\kappa\equiv 2\ell\nu\,,$ (7)

where $\ell=L/\sqrt{T}\ll 1$. The quantity $\nu=U\sqrt{T}/W$ is the re-scaled
regularized local energy density, while $\kappa=2\ell\nu$ is the fraction of
the total energy observed inside the interval $|x|<L$.

The (rescaled) optimal path of the process conditioned on $U$ is described by
the well-known MFT equations for the KMP model JonaLasinioreview ; DG2009b .
They can be recast in the following symmetric form MS2013 ; BSM2022a :

$\displaystyle\partial_{t}u$ $\displaystyle=$
$\displaystyle\partial_{x}(\partial_{x}u+2u^{2}v)\,,$ (8)
$\displaystyle\partial_{t}v$ $\displaystyle=$
$\displaystyle\partial_{x}(-\partial_{x}v+2uv^{2})\,.$ (9)

Equations (8) and (9) follow from a minimization of the action functional of a
proper path integral, constrained by Eq. (7), with respect to variations of
$u(x,t)$ JonaLasinioreview ; DG2009b ; MS2013 ; BSM2022a . The minimization
also yields, aside from Eqs. (8) and (9), a boundary condition at the (re-
scaled) final time,

$v(x,t=1)=\lambda\left[\delta(x-\ell)-\delta(x+\ell)\right]\,,$ (10)

where $\lambda$ is a Lagrange multiplier, to be ultimately expressed through
$\nu$ and $\ell$ with the help of Eq. (7) SmithMS . Importantly, the MFT
equations and boundary conditions for this problem obey the mirror symmetry
and antisymmetry relations

$u(-x,t)=u(x,t)\quad\text{and}\quad v(-x,t)=-v(x,t)\,.$ (11)

Once $u(x,t)$ and $v(x,t)$ have been found, one can calculate the rescaled
action $s$ DG2009b ; MS2013

$\displaystyle
s=\int_{0}^{1}dt\int_{-\infty}^{\infty}dx\,u^{2}(x,t)\,v^{2}(x,t)=2\int_{0}^{1}dt\int_{0}^{\infty}dx\,u^{2}(x,t)\,v^{2}(x,t)\,.$
(12)

It is often simpler, however, to determine the action by using the “shortcut
relation”

$\frac{ds}{d\kappa}=\lambda(\kappa)\,,$ (13)

which follows from $\kappa$ and $\lambda$ being conjugate variables, see e.g.
Ref. shortcut . The action gives the desired probability distribution
$\mathcal{P}$ up to a sub-leading, pre-exponential factor that we do not
attempt to evaluate:

$\ln{\mathcal{P}}(U,T,\ell)\simeq-\sqrt{T}\,s\left(\nu,\ell\right).$ (14)

Since $\sqrt{T}\gg 1$, Eq. (14) clearly has a large-deviation structure, with
the re-scaled action $s\left(\nu,\ell\right)$ playing the role of a rate
function.

The main objective of this work is to determine the large-deviation function
$s\left(\nu,\ell\right)$ in the leading order in the small regularization
parameter $\ell$ but for arbitrary $\nu$. In Sec. II we describe typical,
small fluctuations of $\nu$ around its expected value
$\bar{\nu}=\bar{U}\sqrt{T}/W$. Here one can use the Lagrange multiplier
$\lambda$ as a small expansion parameter KrMe . We also present in Sec. II a
free-energy argument that, as we argue, applies in a broad range of the
densities $\nu$. In Sec. III we apply the Inverse Scattering Method to obtain
important relations which enable us to determine the complete rate function
$s\left(\nu,\ell\right)$, to justify the free-energy argument and establish
its applicability limits, and to find additional asymptotics of the rate
function. In particular, we show that, in the limit of $\ell\to 0$, the rate
function expressed in terms of $\kappa=2\ell\nu$ acquires a universal
($\ell$-independent) form. We also present two instructive examples of the
optimal configurations $u(x,1)$ at different $\lambda$ (or $\nu$). Section IV
briefly summarizes and discusses our results.

##  II Linear theory and free-energy argument

Typical fluctuations of $\nu$ around its expected value
$\bar{\nu}=\bar{U}\sqrt{T}/W$ are Gaussian. Here one can linearize Eqs. (8)
and (9) with respect to $|\lambda|\ll 1$ KrMe . Then Eq. (9) becomes simply
$\partial_{t}v=-\partial_{x}^{2}v$. This anti-diffusion equation can be solved
backward in time with the “initial” condition (10). The solution, for
arbitrary $\ell$, is

$v(x,t)=\frac{\lambda}{\sqrt{4\pi(1-t)}}\,\left[e^{-\frac{(x-\ell)^{2}}{4(1-t)}}-e^{-\frac{(x+\ell)^{2}}{4(1-t)}}\right]\,.$
(15)

The action $s$ in the leading (second) order in $\lambda$ can be obtained from
Eq. (12) by using the solution (15) for $v(x,t)$ and setting
$u(x,t)=\bar{u}(x,t)$ from (a re-scaled) Eq. (4). Evaluating the double
integral, we arrive at the following expression:

$s(\lambda,\ell)=\frac{\lambda^{2}}{4\sqrt{2\pi}}\left[e^{-\frac{\ell^{2}}{2}}-\text{erfc}\left(\frac{\ell}{\sqrt{2}}\right)\right]\,.$
(16)

Then, using the shortcut relation (13), we can express the re-scaled action in
terms of $\nu$:

$s(\nu,\ell)=\frac{4\sqrt{2\pi}\,\ell^{2}\,e^{\frac{\ell
2}{2}}}{1-e^{\frac{\ell^{2}}{2}}\text{erfc}\left(\frac{\ell}{\sqrt{2}}\right)}\,\left(\nu-\bar{\nu}\right)^{2}\,,$
(17)

where

$\bar{\nu}\equiv\frac{\bar{U}\sqrt{T}}{W}=\frac{1}{2\ell}\,\text{erf}\left(\frac{\ell}{2}\right)\,,$
(18)

and the validity of the linear theory requires that
$|\nu-\bar{\nu}|\ll\bar{\nu}$.

For $\ell\ll 1$, that is at sufficiently long time at fixed $L$, Eqs. (17) and
(18) simplify to

$s(\nu,\ell)\simeq 4\pi\ell\left(\nu-\bar{\nu}\right)^{2}$ (19)

and

$\bar{\nu}\simeq\frac{1}{\sqrt{4\pi}}\,,$ (20)

respectively. The variance of fluctuations of $\nu$ is equal to
$1/(8\pi\ell)$. It diverges in the limit of $\ell\to 0$, exhibiting the
ultraviolet catastrophe SmithMS . The $1/\ell$ scaling of the variance is to
be expected. Indeed, the final re-scaled energy density inside the small
interval $|x|<\ell$ is already almost constant and equal to $1/(8\pi\ell)$. In
its turn, the variance of $\nu$ scales as one over the energy content of this
interval.

For future reference, the small-$\ell$ asymptotic of the action (16) in terms
of $\lambda$ is

$s(\lambda,\ell\ll 1)=\frac{\ell\,\lambda^{2}}{4\pi}\,.$ (21)

Alongside with the shortcut relation (13), this again leads to the
small-$\ell$ asymptotic (19).

When $\ell$ is small, the linear theory predicts a flat optimal density
profile at the observation time in the “interior region” $|x|\ll\ell$:

$u(|x|<\ell,t=1)=\frac{1}{\sqrt{4\pi}}+\frac{\lambda}{4\pi}+\dots\,,$ (22)

where the ellipsis denotes sub-leading corrections in $\ell\ll 1$. By assuming
a flat density profile $u(|x|<\ell,1)=\nu$ from the start, one can obtain Eq.
(22) from Eqs. (13), (19) and (20).

Importantly, at $\ell\ll 1$, the gas in the small interior region $|x|<\ell$
has a sufficient time to reach thermal equilibrium SmithMS . It is not
surprising, therefore, that the Gaussian asymptotic (21), and hence (19), for
the action also follows from the Boltzmann-Gibbs free energy difference
DG2009b ; KrMe

$s\simeq\int_{-\ell}^{\ell}dx\left[F(u(x,1))-F(\bar{u})-F^{\prime}(\bar{u})(u(x,1)-\bar{u})\right]\,,$
(23)

where $F(u)=-\ln u$ is the equilibrium free energy density of the KMP gas, see
e.g. Refs. DG2009b ; KrMe . To see it, one should set in Eq. (23)
$\bar{u}=\bar{\nu}=1/\sqrt{4\pi}$, use Eq. (22) for $u(|x|<\ell,1)$ and expand
the integrand in powers of $\lambda$ up to, and including, a quadratic term.

As we will show in Sec. III.1, at $\ell\ll 1$ Eq. (23) actually holds well
beyond the linear theory, once the density at the final time $t=1$ is
approximately uniform in the interior region:
$u(|x|<\ell,1)=\text{const}=\nu$. In this case Eq. (23) predicts

$s_{\text{equil}}(\nu,\ell)\simeq
2\ell\left(\frac{\nu}{\bar{\nu}}-\ln\frac{\nu}{\bar{\nu}}-1\right)\,.$ (24)

Notice that the rate function (24) diverges logarithmically at $\nu\to 0$.
This important property is characteristic of the KMP gas, and it can be traced
down to the nature of its microscopic model, which deals with a continuous
energy variable rather than discrete particles.

##  III Solution by the Inverse Scattering Method

As was noticed in Ref. BSM2022a , the MFT equations (8) and (9) formally
coincide with the derivative nonlinear Schrödinger equation (DNLSE) in
imaginary space and time. Application of the Inverse Scattering Method (ISM)
to the DNLSE was pioneered by Kaup and Newell KaupNewell  for initial-value
problems. Adaptation of the method to boundary-value problems in time has been
described in detail in several papers BSM2022a ; BSM2022b ; KLD2023 ; BM2024 .
Nevertheless, we outline here the main steps of the method, as it applies to
the current problem, forgoing a derivation of Eq. (28) below, and skipping
some of the actual calculations, as they can be easily recovered. Following
Kaup and Newell KaupNewell , one starts with the auxiliary scattering problem
for the field $\bm{\psi}(x,t;k)$,

$\displaystyle\partial_{x}\bm{\psi}(x,t;k)=V(x,t;k)\bm{\psi}(x,t;k)\,,$ (25)

where

$\displaystyle V(x,t;k)=\begin{pmatrix}-\frac{\zeta^{2}}{2}&-iv(x,t)\zeta\\\
-iu(x,t)\zeta&\frac{\zeta^{2}}{2}\\\ \end{pmatrix}$ (26)

and $\zeta=\sqrt{ik}$. The variable $k$ is a wave number describing the plane-
wave boundary conditions at $x\to\infty$ for $\bm{\psi}(x,t;k)$, while the
$t$-dependence comes into play in the form of dependence of $V$ on $t$ through
$u$ and $v$. Namely, one assumes $\bm{\psi}(x,t;k)=\begin{pmatrix}\alpha
e^{-ikx}\\\ \beta e^{ikx}\\\ \end{pmatrix}$ at $x\to-\infty$. Then, using the
linearity of the problem and the form of the solution at infinity, where $u$
and $v$ vanish, one obtains the following asymptotic behavior at
$x\to+\infty$:

$\displaystyle\bm{\psi}(x,t;k)=\begin{pmatrix}\alpha
a(k;t)e^{-ikx}+\beta\tilde{b}(k;t)e^{-ikx}\\\ \alpha
b(k;t)e^{ikx}+\beta\tilde{a}(k;t)e^{ikx}\\\ \end{pmatrix}\,.$ (27)

The integrability of the problem manifests itself in the fact that the time
evolution of the parameters $a,$ $\tilde{a}$, $b$ and $\tilde{b}$ is very
simple:

$\displaystyle a(k;t)=a(k;0),\quad\tilde{a}(k;t)=\tilde{a}(k,0),\quad
e^{k^{2}t}b(k,t)=b(k,0),\quad\text{and}\quad\tilde{b}(k,t)=\tilde{b}(k,0)e^{k^{2}t}\,.$
(28)

These relations, alongside with the fact that $a(k;t)$ is analytic in the
upper half $k$-plane, while $\tilde{a}(k;t)$ is analytic in the lower half
plane KaupNewell , allows one to find explicit expressions for $v(x,0)$ and
$u(x,1)$, as we shall see below.

Now we proceed to calculating $\tilde{b}(k;0)$ and $\tilde{a}(k;0)$ by solving
the scattering problem, Eq. (25), at time $t=0$ setting $\alpha=0$ and
$\beta=1$. The solution of the scattering problem is greatly aided by the fact
that $u(x,0)=\delta(x)$, see Eq. (10). We obtain

$\displaystyle\tilde{b}(k;0)=-i\zeta\left[Q(k)-ikQ_{+}(k)Q_{-}(k)\right]\quad\text{and}\quad\tilde{a}(k;0)=1-ikQ_{-}(k)\,,$
(29)

where

$\displaystyle
Q_{+}(k)=\int_{0}^{\infty}e^{\zeta^{2}x^{\prime}}v(x^{\prime})dx^{\prime},\quad
Q_{-}(k)=\int_{-\infty}^{0}e^{\zeta^{2}x^{\prime}}v(x^{\prime})dx^{\prime}\quad\text{and}\quad
Q(k)=Q_{+}(k)+Q_{-}(k)\,.$ (30)

A similar exercise can be performed at $t=1$, this time making use of the
simple form of $v(x,1)$ given by Eq. (10) to solve the scattering problem, Eq.
(25). However, in order to carry out this calculation, one must first make
some assumptions about the behavior, in the limit of $\ell\to 0$ that we are
interested in, of $u(x,1)$ in the two natural regions that space is divided
into due to the form of $v(x,1)$, namely the interior region $|x|<\ell$ and
the exterior region $|x|>\ell$. We make the following assumption, to be
validated a posteriori. We assume that, as $\ell\to 0$, the quantity
$u(x,1)\ell$ in the interior region $|x|<\ell$ converges to a certain finite
$x$-independent value $u_{0}\ell$. (We will ultimately identify the quantity
$u_{0}$ with $\nu$). Because of the shocks of $u(x,1)$ at $|x|=\ell$, the
value of $u_{0}$ is different from the limiting value of $u(x,1)$ as
$x\to\pm(\ell+0^{+})$. We denote the latter by $u_{\pm}$. Let us also denote
by $v_{\pm}$ the limiting values of $v(x,0)$ as $x\to 0^{\pm}$. We show in the
following, the quantities $v_{\pm}$ actually vanish, but we do not assume it a
priori. We summarize these definitions here:

$\displaystyle u_{0}=\ell^{-1}\lim_{\ell\to 0}\ell u(x,1)\mbox{ for
}|x|<\ell,\quad u_{\pm}=u[x\pm(\ell+0^{\pm}),1],\quad v_{\pm}=v(x\pm
0^{+},0).$ (31)

We also remind the reader that $u(x,1)$ is mirror symmetric with respect to
$x$, while $v(x,0)$ is antisymmetric, see Eq. (11).

Performing the scattering transform at $t=1$ and using the small parameter
$\ell\ll 1$, we arrive at the following relations:

$\displaystyle\tilde{b}(k;1)=2\lambda(\lambda
u_{0}+1)\zeta\sin(k\ell)\simeq-2i\lambda\ell\zeta^{3}(\lambda u_{0}+1)\,,$
(32) $\displaystyle\tilde{a}(k;1)=1-2\lambda\ell iku_{0}-2\lambda\ell
k^{2}(1+\lambda u_{0})\int_{\ell}^{\infty}e^{-ikx}u(x,1)dx\,.$ (33)

Now, using Eq. (28) to compare of $\tilde{a}(k;t)$ at $t=0$ and $t=1$, we
obtain

$\displaystyle 2\lambda\ell u_{0}-2\lambda\ell ik(1+\lambda
u_{0})\int_{\ell}^{\infty}e^{-ikx}u(x,1)dx=-\int_{-\infty}^{0}e^{ikx}v(x,0)dx\,.$
(34)

Setting $k=0$, we obtain

$u_{0}=-\frac{1}{2\lambda\ell}\int_{-\infty}^{0}v(x,0)dx\equiv\frac{1}{2\lambda\ell}\int_{0}^{\infty}v(x,0)dx\,.$
(35)

Sending $k$ to $\infty$, we observe that the dominant contribution to the
integral over $x$ on the l.h.s. comes from a narrow region of width $O(1/k)$
near $x=\ell$. In this narrow region one can replace $u(x,1)$ by $u_{0}$. The
integral over $x$ on the r.h.s. vanishes in the leading order as $k\to\infty$,
and we obtain

$u_{0}=u_{+}(1+\lambda u_{0})\,.$ (36)

Now we rewrite $-ike^{-ikx}$ on the l.h.s of Eq. (34) as
$\partial_{x}e^{-ikx}$, perform integration by parts, and send $\ell$ to $0$
in the leading order in small $\ell$. Then we use Eq. (36), observe a
cancellation of two terms and perform inverse Fourier transform of the both
sides of the equation. We obtain the following relation between $u(x,1)$ and
$v(x,0)$:

$\displaystyle v(x,0)=-2\lambda\ell(1+\lambda u_{0})\partial_{x}u(x,1),$ (37)

In a similar way, by comparing $\tilde{b}$ at $t=0$ and $t=1$, one obtains the
relation

$\displaystyle-i\zeta\left(Q-ikQ_{+}Q_{-}\right)=-i\zeta^{3}2\lambda\ell(1+\lambda
u_{0})\,,$ (38)

which can be recast in the following form:

$\displaystyle\left(1-ikQ_{+}\right)\left(1-ikQ_{-}\right)=1+2\lambda\ell(1+\lambda
u_{0})k^{2}e^{-k^{2}}\,.$ (39)

Noticing that the quantities $1-ikcQ_{\pm}$ are analytic in the upper and
lower half $k$-planes respectively, we recognize this equation as the
decomposition of the function on the right hand side into a product of
functions analytic in the upper and lower half planes, respectively. It is
then natural to take the logarithm of this equation. This is facilitated by
the following definition of $M_{\pm}$:

$\displaystyle 1-ikQ_{\pm}(k)=(1\pm v_{\pm})e^{M_{\pm}},$ (40)

where one recognizes in $M_{\pm}$ the logarithm of $1-ikQ_{\pm},$
respectively, but up to a constant, which is given by $(1+v_{\pm}$), where
$v_{\pm}$ are defined in Eq. (31). These factors allow one to define $M_{\pm}$
as functions which go to $0$ as $k\to\infty$. Then, taking this limit on both
sides of Eq. (40), one can see that on the left hand side one obtains $1\pm
v_{\pm}$ as a consequence of the definition of $Q_{\pm}(k)$, Eq. (30). The
right hand side is then decorated by the factors $1\pm v_{\pm}$ in order to
match this asymptotic.

The decomposition of the logarithm of the right hand side of Eq. (39) into
$M_{\pm}$, which in turn is the logarithm (up to a constant) of $1-ikQ_{\pm},$
is performed in the usual way to give the following result:

$\displaystyle
M_{\pm}=\pm\int_{-\infty}^{\infty}\frac{\ln\left(1+2\lambda\ell\left(1+\lambda
u_{0}\right)k^{2}e^{-k^{\prime 2}}\right)}{k-k^{\prime}\mp
i0^{+}}\frac{dk^{\prime}}{2\pi i}$ (41)

which yields an expression for $Q_{\pm}$:

$\displaystyle ikQ_{\pm}(k)=1-\left(1\pm
v_{\pm}\right)\exp\left[\pm\int_{-\infty}^{\infty}\frac{\ln\left(1+2\lambda\ell\left(1-\lambda
u_{0}\right)k^{2}e^{-k^{2}}\right)}{k^{\prime}-k\mp
i0^{+}}\frac{dk^{\prime}}{2\pi i}\right].$ (42)

Since $Q_{\pm}$ is the Fourier transform of $\theta(\pm x)v(x,0)$,
respectively, this equation alongside with Eq. (37) gives $v(x,0)$ and
$u(x,1)$, provided that $v_{\pm}$ and $u_{0}$ are found.

The quantities $v_{\pm}$ can be found from a demand that $Q_{\pm}(k)$ be
regular at the origin. This is equivalent to a demand that $u(x,1)$ tend to
zero at infinity, which in our problem is satisfied automatically. Setting
$k=0$ in Eq. (42), one can see that the quantities $v_{\pm}$, defined in Eq.
(31), vanish, and the factors $1+v_{\pm}$ in Eq. (42) are both equal to $1$.
Combining Eqs. (30), (42) and (35), we arrive at a transcendental equation for
$u_{0}=\nu$:

$\displaystyle
4\pi\lambda\ell\nu=\int_{-\infty}^{\infty}\frac{\ln\left[1+2\lambda\ell\left(1+\lambda\nu\right)k^{2}e^{-k^{2}}\right]}{k^{2}}dk\,.$
(43)

Equations (42) (with $v_{\pm}=0$) and (43) yield $v(x,0)$, while the
relationship (37) between $v(x,0)$ and $\partial_{x}u(x,1)$, allows one to
find $u(x,1)$ as well. In fact, Eq. (43), combined with the “shortcut
relation” (13), suffice for the purpose of calculating the action
$s(\nu,\ell)$ which serves as the rate function of the probability density
$\mathcal{P}(U,T,\ell)$, see Eq. (14).

###  III.1 Rate function and its asymptotics

For general $\lambda,$ Eq. (43) for $\nu$ can only be solved numerically. As
an example, the left panel of Fig. 1 shows a fragment of the resulting
dependence of $\nu$ upon $\lambda$ for $\ell=0.01$. The point $\lambda=0$
corresponds to $\nu=\bar{\nu}=1/\sqrt{4\pi}$, as to be expected. The right
panel of Fig. 1 shows the rate function $s(\nu)$ obtained from Eq. (13) by
integrating numerically the inverse function $\lambda(\nu)$. The inset
compares this $s(\nu)$ with predictions from the linear theory, Eq. (19), and
from the free-energy difference, Eq. (24). As one can see, the Gaussian
asymptotic, predicted by the linear theory, applies only in a narrow region of
$\nu$ around $\bar{\nu}$, whereas the free-energy difference captures $s(\nu)$
in a much broader range of $\nu$.

Figure 1: Left panel: $\nu=\nu(\lambda)$, found by numerically solving Eq.
(43) for $\ell=0.01$ and different $\lambda$. Right panel: the rate function
$s$ vs. $\nu$ for this $\ell$ (the solid line), the linear theory prediction
(19) (the dashed line) and the prediction from the free-energy difference (24)
(the dot-dashed line). The inset shows a blowup of the interval
$0<\nu<\bar{\nu}$.

To understand the formal reason behind the latter finding, we notice that,
because of the factor $e^{-k^{2}}$ of the integrand in Eq. (43), the
characteristic width of the integration region in $k$ is $O(1)$. Let us assume
(and check a posteriori) that

$2|\lambda\ell\left(1+\lambda\nu\right)|\ll 1\,,$ (44)

and expand the logarithm keeping only the linear term:
$\ln\left[1+2\lambda\ell\left(1+\lambda\nu\right)k^{2}e^{-k^{2}}\right]\simeq
2\lambda\ell\left(1+\lambda\nu\right)k^{2}e^{-k^{2}}$. Evaluating the
elementary integral and solving the resulting simple algebraic equation for
$\nu$, we obtain

$\nu=\frac{1}{\lambda_{c}-\lambda}\,,\quad\text{where}\quad\lambda_{c}=\sqrt{4\pi}\equiv
1/\bar{\nu}\,.$ (45)

Integrating Eq. (13) with $\kappa=2\ell\nu=2\ell/(\lambda_{c}-\lambda)$, we
reproduce the free-energy prediction (24). As we already discussed, the
Gaussian asymptotic (19), predicted by linear theory, is a particular limit of
Eq. (24), only valid in the vicinity of $\nu=\bar{\nu}$. To obtain this
asymptotic directly from Eq. (43), one should expand the integrand in the
powers of $\lambda$ up to and including the second order, and keep only the
leading (first-order) term in $\ell\ll 1$.

Importantly, for Eq. (45) to make sense, $\lambda$ must obey the condition

$-\infty<\lambda<\lambda_{c}\,.$ (46)

In addition, the assumed strong inequality (44) can be justified only if
$\lambda$ is not too close to $\lambda_{c}$: $\lambda_{c}-\lambda\gg\ell$. As
a result, the free-energy prediction (24) for the rate function applies for
all $\nu$ much smaller than $O(1/\ell)\gg 1$.

What happens at $\nu\gtrsim O(1/\ell)$? Here the rate function can be
determined numerically, as shown in Fig. 1. However, an important
simplification occurs as $\ell\to 0$. As can be checked a posteriori, here the
term $\lambda\nu$ inside the parentheses in Eq. (43), is much larger than $1$.
Neglecting this $1$ in the zeroth order in $\ell$ at fixed $\kappa=2\ell\nu$,
one arrives at a universal equation for $\kappa=2\ell\nu$ versus $\lambda$:

$2\pi\lambda\kappa=\int_{-\infty}^{\infty}\frac{\ln\left(1+\lambda^{2}\kappa
k^{2}e^{-k^{2}}\right)}{k^{2}}dk\,.$ (47)

The resulting function $\kappa(\lambda)$, obtained numerically, is shown on
the left panel of Fig. 2. In this limit $\kappa(\lambda)$ vanishes at
$\lambda<\lambda_{c}$, and it undergoes a tangent bifurcation at
$\lambda=\lambda_{c}$. The right panel shows the resulting universal rate
function $s_{>}(\kappa)$ alongside with its two asymptotics: $s_{>}(\kappa\to
0)\simeq\sqrt{4\pi}\,\kappa$, and the asymptotic at $\kappa\to 1$.
Importantly, the $\kappa\to 0$ asymptotic matches the $\nu\gg\bar{\nu}$
asymptotic of the equilibrium rate function, which is described by the first
term in the r.h.s. of Eq. (24).

The asymptotic at $\kappa\to 1$ can be expressed in terms of product log
(Lambert $W$) function, very similarly to how it has been done in Ref.
BSM2022a . Alternatively, it can be presented in a parametric form as follows:

$\kappa(\lambda\to\infty)\simeq
1-\frac{6\sqrt{\ln\lambda}}{\pi\lambda}\,,\quad
s_{>}(\lambda\to\infty)\simeq\frac{4(\ln\lambda)^{3/2}}{\pi}\,.$ (48)

Figure 2: Left panel: $\kappa=2\ell\nu$ versus $\lambda$, found by numerically
solving the universal equation (45). The inset focuses on the region close to
the tangent bifurcation point $\lambda=\lambda_{c}=\sqrt{4\pi}$. Right panel:
the rate function $s_{>}(\kappa)$ in terms of the energy fraction $\kappa$
alongside with its small-$\kappa$ asymptotic $s_{>}(\kappa\to
0)\simeq\sqrt{4\pi}\,\kappa$, and the large-$\kappa$ asymptotic (48).

When $\ell$ is small but finite, the tangent bifurcation at
$\lambda=\lambda_{c}$ is smoothed out, and $\kappa$ is non-zero even for
$\lambda<\lambda_{c}$. However, as $\ell$ is reduced, the functions
$\kappa(\lambda,\ell)$, calculated from Eq. (43) for different $\ell$,
converge to the universal curve $\kappa(\lambda)$. This convergence in evident
in Fig. 3: at $\ell=10^{-4}$ the function $\kappa(\lambda,\ell)$ is already
almost indistinguishable from the universal curve shown in the inset of the
left panel of Fig. 2.

Figure 3: Convergence of $\kappa(\lambda,\ell)$ to the universal function
$\kappa(\lambda)$ for $\ell=10^{-2}$, $3\cdot 10^{-3}$ and $10^{-4}$ (from top
to bottom).

Now we are in a position to examine the validity of our analytical results.
The main assumption we made in order to arrive at Eqs. (32) and (33) is that,
to a leading order in $\ell$, the integral $\int_{-\ell}^{\ell}e^{-ikx}u(x,1)$
is independent of $k$ in the relevant range of $k$ in the problem, and
therefore it can be approximated by $2\nu\ell$. This requires that the
characteristic length scale of $u(x,1)$, which is $O(\ell)$, be much smaller
than the relevant $1/k$. The latter is determined by the effective width of
the integration region in Eq. (43). For $-\infty<\lambda<\lambda_{c}$, $\nu$
is given by Eq. (45). As one can check, in this case the relevant range of $k$
is $O(1)$, and the strong inequality $\ell\ll 1$ suffices.

For $\lambda\gtrsim\lambda_{c}$ an additional condition appears. This is
because in this case we obtain $\nu\ell\simeq 1$. As a result, the effective
width of the integration region over $k$ in Eq. (43) is
$O(\sqrt{\ln\lambda^{2}})$. This quantity should be much smaller than
$1/\ell$:

$\ell\sqrt{\ln\lambda^{2}}\ll 1\,.$ (49)

This condition invalidates the rate function, that we found here, when
$\kappa$ becomes too close to its (unreachable) maximum possible value $1$.

###  III.2 Optimal density configurations at $t=1$

Figure 4: Solid lines: $u(x,1)$ for $\lambda=10$ (left panel) and $-30$ (right
panel) as described by the ISM for $\ell=0.05$. Dashed lines: the expected
density profile $(4\pi)^{-1/2}e^{-x^{2}/4}$. Note the different vertical
scales.

As we have seen, the ISM also predicts the optimal density profile $u(x,1)$
and the optimal profile of the conjugate field $v(x,0)$, see Eqs. (37), (42)
and (43). Two examples of the optimal density profiles at $t=1$ for
$\lambda=10$ and $\lambda=-30$ are shown in Fig. 4. For comparison, also shown
is the expected, that is mean-field, density profile
$e^{-x^{2}/4}/\sqrt{4\pi}$.

##  IV Summary and Discussion

By combining the MFT and the ISM, we calculated the long-time large-deviation
statistics of the locally-averaged energy density in a freely expanding
Kipnis-Marchioro-Presutti lattice gas. We showed that the corresponding rate
function exhibits two distinct regions. One region corresponds to relatively
small local densities and is described by the equilibrium free-energy
argument. The other region, associated with large densities, is fundamentally
non-equilibrium. As time increases at a fixed $L$, the rate function converges
to a universal form when expressed in terms of the energy content of the
interval $|x|<\ell$ at the final time.

An attempt to solve this problem without relying on the strong inequality
$\ell\ll 1$ would be an interesting endeavor. Such an extension is not
required in the context of regularization of the single-point density. On the
other hand, it would make an interesting connection to the statistics of
accumulated current of matter or energy in a specified region of space DG2009a
; DG2009b . There is also an interesting formal analogy between the setting
with arbitrary $\ell$ and the problem of finding the joint distribution of two
accumulated currents in the Simple Symmetric Exclusion Process. The latter
problem has been recently formulated in terms of the ISM (for the nonlinear
Schrödinger equation rather than for the DNLSE), and a possible approach to
the solution has been explored GRB2024 .

In conclusion, a growing list of field-theoretical large-deviation problems
have been successfully solved with the aid of the ISM in the last 3 years
KLD2021 ; KLD2022 ; BSM2022a ; Mallick2022 ; BSM2022b ; KLD2023 ; BM2024 .
However, this work appears to be the first instance where the ISM has been
combined with a perturbation approach exploiting a small parameter (in this
case $\ell$), unrelated to a Lagrange multiplier. We believe that this
approach holds significant promise.

### Acknowledgments

We are very grateful to N. R. Smith for a critical reading of the paper and
insightful comments. We also thank A. Grabsch and O. Bénichou for attracting
our attention to their recent work GRB2024 . This research was supported by
the US-Israel Binational Science Foundation through Grant No. 2020193 (E.B.)
and by the Israel Science Foundation through Grant 1499/20 (B.M.).

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