# On the rate of convergence of the shape of Young diagrams associated with
random words

Clément Deslandes111C.M.A.P. École Polytechnique, Palaiseau, 91120, France &
Georgia Institute of Technology, Atlanta, GA, 30332, USA
(clement.deslandes@poytechnique.edu). Christian Houdré222School of
Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA
(houdre@math.gatech.edu). 333Research supported in part by the grant $\sharp
524678$ from the Simons Foundation.
Keywords: Random Words, Longest Increasing Subsequences, Young Diagrams, RSK
Correspondence, KMT Approximation, Kiefer Processes, Weak Convergence, Last
Passage Percolation, Random Matrices.
MSC 2010: 05A05, 60C05, 60F05.

###### Abstract

We revisit, beyond the uniform case, some aspects of the convergence of the
cumulative shape of the RSK Young diagrams associated with random words,
obtaining rates of convergence in Kolmogorov’s distance. Since the length of
the top row of the diagrams is the length of the longest increasing
subsequences of the word, a corresponding rate result follows. This is then
extended to the length of the longest common and increasing subsequences in
two or more random words.

##  1 Introduction

Let $X_{1},\dots,X_{n}$ be i.i.d. random variables with values in a finite
alphabet $\\{1,\dots,m\\}$ and with probability mass function given by
$p_{1},\dots,p_{m}$. For $i\in\\{1,\dots,m\\}$ and $j\in\\{0,\dots,n\\}$, and
with elements of the notation in [5, 2, 3], let
$N^{i}_{j}=\sum_{l=1}^{j}\mathds{1}_{X_{l}=i}$ be the number of letters $i$
within the first $j$ letters, and for $t\in[0,1]$, let

$\widetilde{B^{n}_{i}}(t)=\frac{N^{i}_{\lfloor tn\rfloor}-p_{i}\lfloor
tn\rfloor}{\sigma_{i}\sqrt{n}},$ (1.1)

where $\sigma_{i}=\sqrt{p_{i}(1-p_{i})}$. Let
$\Lambda=\\{\lambda\in[0,1]^{m}:\lambda_{1}+\dots+\lambda_{m}=1\\}$, and for
$\lambda\in\Lambda$, let

$\widetilde{V^{n}_{i}}(\lambda)=\sigma_{i}\left(\widetilde{B^{n}_{i}}(\lambda_{1}+\dots+\lambda_{i})-\widetilde{B^{n}_{i}}(\lambda_{1}+\dots+\lambda_{i-1})\right),$
(1.2)

$i\in\\{1,\dots,m\\}$ with for $i=1$, the convention that
$\widetilde{B^{n}_{i}}(\lambda_{1}+\dots+\lambda_{i-1})=0$.

Hence, $LI_{n}$, the length of the longest increasing subsequences of
$X_{1}\cdots X_{n}$ is given by:

$LI_{n}=\max_{0=k_{0}\leq k_{1}\leq\dots\leq
k_{m}=n}\sum_{i=1}^{m}\left(N^{i}_{k_{i}}-N^{i}_{k_{i-1}}\right)=\max_{\begin{subarray}{c}\lambda\in\Lambda_{d}\end{subarray}}\sum_{i=1}^{m}\left(np_{i}\lambda_{i}+\sqrt{n}\widetilde{V^{n}_{i}}(\lambda)\right),$
(1.3)

where now
$\Lambda_{d}:=\\{\left({j_{1}}/{n},\dots,{j_{m}}/{n}\right):j_{1},\dots,j_{m}\in\mathbb{N},j_{1}+\dots+j_{m}=n\\}$.

For $\lambda\in\Lambda$ let, finally,

$Z_{n}(\lambda):=\sum_{i=1}^{m}\left(\sqrt{n}(p_{i}-p_{\max})\lambda_{i}+\widetilde{V^{n}_{i}}(\lambda)\right),$
(1.4)

where $p_{\max}=\max_{i=1,\dots,m}p_{i}$, so that

$\frac{LI_{n}-np_{\max}}{\sqrt{n}}=\max_{\begin{subarray}{c}\lambda\in\Lambda_{d}\end{subarray}}Z_{n}(\lambda).$
(1.5)

It is known (see [5, 2, 3] and the references therein) that the limiting
distribution of (1.5) is the distribution of
$\max_{\begin{subarray}{c}\lambda\in\Lambda\end{subarray}}Z^{\prime}_{n}(\lambda)$
where $Z^{\prime}_{n}$ is defined as $Z_{n}$ but with $B^{n}_{i}$, a Brownian
motion, instead of $\widetilde{B^{n}_{i}}$ (as stated more precisely in the
sequel). Note that $Z^{\prime}_{n}$ has the same distribution for all $n$, so
that the limiting distribution above is well defined. Below, our main goal is
to provide a rate of convergence for this result. To do so, the strategy is to
use a KMT approximation to build a coupling: we will define on the same
probability space $\widetilde{B}$ and $B$ that are very close. Then, when the
letters are uniformly distributed, $Z_{n}(\lambda)$ simplifies to
$Z_{n}(\lambda)=\sum_{i=1}^{m}\widetilde{V^{n}_{i}}(\lambda)$, and therefore
it is straightforward to infer from the coupling the rate of convergence
bound. This can also be done for the other lines of the RSK Young diagrams
associated with the word. When the distribution is not uniform, the strategy
is to first approximate
$\max_{\begin{subarray}{c}\lambda\in\Lambda_{d}\end{subarray}}Z_{n}(\lambda)$
by
$\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}_{d}\end{subarray}}Z_{n}(\lambda)$,
where $\Lambda^{\prime}_{d}=\\{\lambda\in\Lambda_{d}:p_{i}\neq
p_{\max}\implies\lambda_{i}=0\\}$, because then for each
$\lambda\in\Lambda^{\prime}_{d}$,
$Z_{n}(\lambda)=\sum_{i=1}^{m}\widetilde{V^{n}_{i}}(\lambda)$, as previously.
These approaches also give the convergence of moments. To finish, the case of
two or more common and increasing subsequences is addressed.

##  2 A coupling via KMT and rates of convergence

To start with, we prove the following key coupling lemma:

######  Lemma 2.1.

Let $\alpha\geq 1$. For every $m\geq 2$, every probability mass function
$p_{1},\dots,p_{m}$, and every $n\geq 2$, there exists a probability space
with $X_{1},\dots,X_{n},\widetilde{B^{n}}$, as above, and $B^{n}$ an
$m$-dimensional Brownian motion with covariance matrix
$\Sigma:=\mathrm{Cov}\left((\mathds{1}_{X_{1}=i}/\sigma_{i})_{1\leq i\leq
m}\right)$, defined on it, such that

$\mathds{P}\left(\sup_{\begin{subarray}{c}i\in\\{1,\dots,m\\}\\\ 0\leq t\leq
1\end{subarray}}\left|\sigma_{i}\widetilde{B^{n}_{i}}(t)-\sigma_{i}B^{n}_{i}(t)\right|\geq
C\frac{\alpha(\log n)^{2}}{\sqrt{n}}\right)\leq\frac{2}{n^{\alpha}},$ (2.1)

where $C$ is a universal constant.

###### Proof.

By the KMT approximation [7] (see also [8] for further details and extensive
references, in particular on Kiefer processes), there exist a probability
space with $(U_{i})_{i\geq 1}$ i.i.d. uniform on $[0,1]$ random variables and
a Kiefer process $(K(s,t))_{\begin{subarray}{c}s\in[0,1]\\\
t\in[0,\infty)\end{subarray}}$ such that for all $x>0$,

$\mathds{P}\left(\sup_{\begin{subarray}{c}0\leq s\leq 1\\\
l\in\\{1,\dots,n\\}\end{subarray}}\left|\frac{1}{\sqrt{n}}\sum_{k=1}^{l}\left(\mathds{1}_{U_{k}\leq
s}-s\right)-\frac{K\left(s,l\right)}{\sqrt{n}}\right|\geq C_{1}\log
n\frac{\log n+x}{\sqrt{n}}\right)\leq\mathrm{e}^{-x},$ (2.2)

where $C_{1}$ is a universal constant (throughout, $C_{2},C_{3},\dots$ are
universal constants). In particular, for $x=\alpha\log n$ one gets

$\mathds{P}\left(\sup_{\begin{subarray}{c}0\leq s\leq 1\\\
l\in\\{1,\dots,n\\}\end{subarray}}\left|\frac{1}{\sqrt{n}}\sum_{k=1}^{l}\left(\mathds{1}_{U_{k}\leq
s}-s\right)-\frac{K\left(s,l\right)}{\sqrt{n}}\right|\geq
2C_{1}\alpha\frac{(\log n)^{2}}{\sqrt{n}}\right)\leq\frac{1}{n^{\alpha}}.$
(2.3)

To replace the discrete parameter $l\in\\{1,\dots,n\\}$ by a continuous one
$l^{\prime}\in[0,n]$, note that

$\displaystyle\mathds{P}\left(\sup_{\begin{subarray}{c}0\leq s\leq 1\\\
l\in\\{1,\dots,n\\}\end{subarray}}\sup_{l^{\prime}\in[l-1,l]}\left|K\left(s,l\right)-K\left(s,l^{\prime}\right)\right|\geq
2C_{1}\alpha(\log n)^{2}\right)$
$\displaystyle\leq\sum_{l=1}^{n}\mathds{P}\left(\sup_{\begin{subarray}{c}0\leq
s\leq 1\\\
l^{\prime}\in[l-1,l]\end{subarray}}\left|K\left(s,l\right)-K\left(s,l^{\prime}\right)\right|\geq
2C_{1}\alpha(\log n)^{2}\right)$ $\displaystyle\leq
n\mathds{P}\left(\sup_{\begin{subarray}{c}0\leq s\leq 1\\\ 0\leq t\leq
1\end{subarray}}\left|K\left(s,t\right)\right|\geq 2C_{1}\alpha(\log
n)^{2}\right)$ $\displaystyle\leq
n\mathds{P}\left(2\sup_{\begin{subarray}{c}0\leq s\leq 1\\\ 0\leq t\leq
1\end{subarray}}\left|W\left(s,t\right)\right|\geq 2C_{1}\alpha(\log
n)^{2}\right),$

where $W$ is a two-dimensional Brownian sheet (using the facts that
$(s,t)\mapsto K(s,l)-K(s,l-t)$ and $(s,t)\mapsto W(s,t)-sW(1,t)$ are Kiefer
processes on $[0,1]^{2}$). From [4, Theorem 3], and if $\Phi$ is the standard
normal cumulative distribution function,

$\mathds{P}\left(\sup_{\begin{subarray}{c}0\leq s\leq 1\\\ 0\leq t\leq
1\end{subarray}}\left|W\left(s,t\right)\right|\geq C_{1}\alpha(\log
n)^{2}\right)\leq 4\Phi(-C_{1}\alpha(\log n)^{2}).$ (2.4)

If $C_{1}$ is large enough, which can be assumed without loss of generality,
for all $\alpha\geq 1$ and all $n\geq 2$, $4\Phi(-C_{1}\alpha(\log
n)^{2})\leq{1}/{n^{1+\alpha}}$. Therefore,

$\mathds{P}\left(\sup_{\begin{subarray}{c}0\leq s\leq 1\\\
l\in\\{1,\dots,n\\}\end{subarray}}\sup_{l^{\prime}\in[l-1,l]}\left|K\left(s,l\right)-K\left(s,l^{\prime}\right)\right|\geq
2C_{1}\alpha(\log
n)^{2}\right)\leq\frac{n}{n^{1+\alpha}}=\frac{1}{n^{\alpha}},$

and using (2.3),

$\mathds{P}\left(\sup_{\begin{subarray}{c}0\leq s\leq 1\\\ 0\leq t\leq
1\end{subarray}}\left|\frac{1}{\sqrt{n}}\sum_{k=1}^{\lfloor
tn\rfloor}\left(\mathds{1}_{U_{k}\leq
s}-s\right)-\frac{K\left(s,tn\right)}{\sqrt{n}}\right|\geq
4C_{1}\alpha\frac{(\log n)^{2}}{\sqrt{n}}\right)\leq\frac{2}{n^{\alpha}}.$
(2.5)

For $i\in\\{1,\dots,n\\}$, let
$X_{i}:=\min_{k\in\\{1,\dots,m\\}}\\{k:U_{i}\leq p_{1}+\dots+p_{k}\\}$.
Clearly, the $X_{i}$ are i.i.d random variables with values in
$\\{1,\dots,m\\}$ and probability mass function $p_{1},\dots,p_{m}$. So, with
the notations above,

$\widetilde{B^{n}_{i}}(t)=\frac{N^{i}_{1,\lfloor tn\rfloor}-p_{i}\lfloor
tn\rfloor}{\sigma_{i}\sqrt{n}}=\frac{\sum_{k=1}^{\lfloor
tn\rfloor}\left(\mathds{1}_{U_{k}\leq
p_{1}+\dots+p_{i}}-(p_{1}+\dots+p_{i})-\mathds{1}_{U_{k}\leq
p_{1}+\dots+p_{i-1}}+p_{1}+\dots+p_{i-1}\right)}{\sigma_{i}\sqrt{n}}.$

For $i\in\\{1,\dots,m\\}$ and $t\in[0,1]$, then

$B^{n}_{i}(t):=\frac{K\left(p_{1}+\dots+p_{i},tn\right)-K\left(p_{1}+\dots+p_{i-1},tn\right)}{\sigma_{i}\sqrt{n}},$

are Brownian motions with covariance matrix
$\Sigma:=\mathrm{Cov}\left((\mathds{1}_{X_{1}=i}/\sigma_{i})_{1\leq i\leq
m}\right)$. Note that

$\sup_{\begin{subarray}{c}i\in\\{1,\dots,m\\}\\\ 0\leq t\leq
1\end{subarray}}\left|\sigma_{i}\widetilde{B^{n}_{i}}(t)-\sigma_{i}B^{n}_{i}(t)\right|\leq
2\sup_{\begin{subarray}{c}0\leq s\leq 1\\\ 0\leq t\leq
1\end{subarray}}\left|\frac{1}{\sqrt{n}}\sum_{k=1}^{\lfloor
tn\rfloor}\left(\mathds{1}_{U_{k}\leq
s}-s\right)-\frac{K\left(s,tn\right)}{\sqrt{n}}\right|,$

and so from (2.5) the following coupling inequality:

$\mathds{P}\left(\sup_{\begin{subarray}{c}i\in\\{1,\dots,m\\}\\\ 0\leq t\leq
1\end{subarray}}\left|\sigma_{i}\widetilde{B^{n}_{i}}(t)-\sigma_{i}B^{n}_{i}(t)\right|\geq
8C_{1}\frac{\alpha(\log n)^{2}}{\sqrt{n}}\right)\leq\frac{2}{n^{\alpha}},$

is valid and this gives the desired result (letting $C=8C_{1}$).

∎

From now on, our setting is the probability space introduced in Lemma 2.1 with
its notation.

To start with, we address the case of uniformly distributed letters and study
the rate of convergence, in Kolmogorov distance, for the cumulative shape of
the RSK Young diagrams associated with the random word.

Let $(R_{k}(n,m))_{1\leq k\leq n}$ be the shape of the RSK Young diagram
associated with the random words $X_{1}\cdots X_{n}$ with uniformly
distributed letters over $\\{1,\dots,m\\}$, and define for $1\leq k\leq m$,
$V_{k}(n,m)=\sum_{l=1}^{k}R_{l}(n,m)$ (so that, for example,
$V_{1}(n,m)=R_{1}(n,m)=LI_{n}$). From [5, Corollary 3.1], for properly defined
$I_{k,m}$ (keeping the notations of [5]), $(V_{k}(n,m))_{1\leq k\leq n}$ is
such that:

$\left(\frac{V_{k}(n,m)-kn/m}{\sqrt{n/m}}\right)_{1\leq k\leq
m}\xRightarrow[n\to\infty]{}\left(\max_{\mathbf{t}\in
I_{k,m}}\sum_{j=1}^{k}\sum_{l=j}^{m-k+j}\sqrt{(m-1)/m}\left(B_{l}(t_{j,l})-B_{l}(t_{j,l-1})\right)\right)_{1\leq
k\leq m},$ (2.6)

where the convergence is in distribution, and $B$ is a $m$-dimensional
Brownian motion with covariance matrix having diagonal terms equal to $1$ and
off-diagonal terms equal to $-1/(m-1)$, i.e., the covariance matrix $\Sigma$
of Lemma 2.1. Recall finally that the limiting law in (2.6) is the spectra of
a traceless $m\times m$ GUE matrix.

To simplify notations, let further denote $\left(\max_{\mathbf{t}\in
I_{k,m}}\sum_{j=1}^{k}\sum_{l=j}^{m-k+j}\sqrt{(m-1)/m}\left(B^{n}_{l}(t_{j,l})-B^{n}_{l}(t_{j,l-1})\right)\right)_{1\leq
k\leq m}$ by $(J_{1,m},\dots,J_{m,m})$ and
$\left(({V_{k}(n,m)-kn/m})/{\sqrt{n/m}}\right)_{1\leq k\leq m}$ by
$(T_{1,m},\dots,T_{m,m})$.

######  Theorem 2.2.

For every $n,m\geq 2$ and $1\leq k\leq m$,

$\sup_{x\in\mathbb{R}}\left|\mathds{P}\left(T_{k,m}\geq
x\right)-\mathds{P}\left(J_{k,m}\geq x\right)\right|\leq C(m)\frac{(\log
n)^{2}}{\sqrt{n}},$

where $C(m)$ is a constant only depending on $m$.

###### Proof.

As shown in [5, (3.7)],

$T_{k,m}=\max_{\mathbf{t}\in
I_{k,m}}\sum_{j=1}^{k}\sum_{l=j}^{m-k+j}\sqrt{(m-1)/m}\left(\widetilde{B^{n}_{l}}(t_{j,l})-\widetilde{B^{n}_{l}}(t_{j,l-1})\right),$
(2.7)

so applying (2.1), since $\left|T_{k,m}-J_{k,m}\right|\leq
2k(m-k+1)\sqrt{m}\sup_{\begin{subarray}{c}i\in\\{1,\dots,m\\}\\\ 0\leq t\leq
1\end{subarray}}\left|\sigma_{i}\widetilde{B^{n}_{i}}(t)-\sigma_{i}B^{n}_{i}(t)\right|$,

$\mathds{P}\left(\left|T_{k,m}-J_{k,m}\right|\geq
2k(m-k+1)\sqrt{m}C\alpha\frac{(\log
n)^{2}}{\sqrt{n}}\right)\leq\frac{2}{n^{\alpha}}.$ (2.8)

Recall next that with $\Theta_{k}:\mathbb{R}^{k}\to\mathbb{R}^{k}$ given by
$(\Theta_{k}(x))_{j}=\sum_{i=1}^{j}x_{i}$, $1\leq j\leq k$, it follows that
$\Theta_{m}^{-1}(J_{1,m},\dots,J_{m,m})$ has the same distribution as the
ordered spectrum of an $m\times m$ traceless GUE matrix. Moreover, by a bound
on the joint density of the eigenvalues (see (3.5) in [6]), there exists
$D(m)>0$ bounding the supremum of the density of $J_{k,m}$ (for $1\leq k\leq
m$).

So, for any $x\in\mathbb{R}$,

$\displaystyle\left|\mathds{P}\left(T_{k,m}\geq
x\right)-\mathds{P}\left(J_{k,m}\geq x\right)\right|$
$\displaystyle\leq\mathds{P}\left(\left|T_{k,m}-J_{k,m}\right|\geq
2k(m-k+1)\sqrt{m}C\alpha\frac{(\log n)^{2}}{\sqrt{n}}\right)$
$\displaystyle\qquad\qquad\qquad+\mathds{P}\left(\left|J_{k,m}-x\right|\leq
2k(m-k+1)\sqrt{m}C\alpha\frac{(\log n)^{2}}{\sqrt{n}}\right)$
$\displaystyle\leq\frac{2}{n^{\alpha}}+D(m)2k(m-k+1)\sqrt{m}C\alpha\frac{(\log
n)^{2}}{\sqrt{n}},$

so $\sup_{x\in\mathbb{R}}\left|\mathds{P}\left(T_{k,m}\geq
x\right)-\mathds{P}\left(J_{k,m}\geq x\right)\right|$ is upper bounded as
stated.

∎

As a corollary, we can also study the speed of convergence of
$T_{.,m}=(T_{k,m})_{1\leq k\leq m}$ towards $J_{.,m}=(J_{k,m})_{1\leq k\leq
m}$, in the Kolmogorov distance, rather than coordinate by coordinate. Just as
before, we have

$\mathds{P}\left(\left\|T_{.,m}-J_{.,m}\right\|_{\infty}:=\max_{1\leq k\leq
m}|T_{.,m}-J_{.,m}|\geq 2m^{5/2}C\alpha\frac{(\log
n)^{2}}{\sqrt{n}}\right)\leq\frac{2}{n^{\alpha}},$ (2.9)

and

$\mathds{P}\left(\max_{1\leq k\leq m}\left|J_{k,m}-x\right|\leq
2m^{5/2}C\alpha\frac{(\log n)^{2}}{\sqrt{n}}\right)\leq
mD(m)2m^{5/2}C\alpha\frac{(\log n)^{2}}{\sqrt{n}},$ (2.10)

so for any $x_{1},\dots,x_{m}\in\mathbb{R}$,

$\displaystyle\left|\mathds{P}\left(\max_{1\leq k\leq m}T_{k,m}\geq
x_{k}\right)-\mathds{P}\left(\max_{1\leq k\leq m}J_{k,m}\geq
x_{k}\right)\right|\leq$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathds{P}\left(\left\|T_{.,m}-J_{.,m}\right\|_{\infty}\geq
2m^{5/2}C\alpha\frac{(\log n)^{2}}{\sqrt{n}}\right)$
$\displaystyle+\mathds{P}\left(\max_{1\leq k\leq m}\left|J_{k,m}-x\right|\leq
2m^{5/2}C\alpha\frac{(\log n)^{2}}{\sqrt{n}}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\leq\frac{2}{n^{\alpha}}+mD(m)2m^{5/2}C\alpha\frac{(\log
n)^{2}}{\sqrt{n}}.$

As in [1], for $m$ no longer fixed, let us consider conditions on the
sequences ($m(n))_{n\geq 1}$ (writing just $m$ for $m(n)$) under which for all
$\varepsilon>0$,

$\mathds{P}\left(\left|\left(T_{1,m}-2\sqrt{m}\right)m^{1/6}-\left(J_{1,m}-2\sqrt{m}\right)m^{1/6}\right|\geq\varepsilon\right)\xrightarrow[n\to\infty]{}0.$
(2.11)

Then, since $\left(J_{1,m}-2\sqrt{m}\right)m^{1/6}$ converges in distribution
to the Tracy–Widom distribution $F_{TW}$, this implies that
$\left(T_{1,m}-2\sqrt{m}\right)m^{1/6}$, that is,
$\left(\left(LI_{n}-n/m\right)/\sqrt{n/m}-2\sqrt{m}\right)m^{1/6}$, converges
to $F_{TW}$ as well.

Applying (2.8) gives

$\mathds{P}\left(\left|\left(T_{1,m}-2\sqrt{m}\right)m^{1/6}-\left(J_{1,m}-2\sqrt{m}\right)m^{1/6}\right|\geq
Cm^{1/6}2k(m-k+1)\sqrt{m}\frac{\alpha(\log
n)^{2}}{\sqrt{n}}\right)\leq\frac{2}{n^{\alpha}},$ (2.12)

so that if $m^{1/6}2k(m-k+1)\sqrt{m}C\alpha{(\log n)^{2}}/{\sqrt{n}}$
converges to zero, that is, if $m=o((n/(\log n)^{4})^{3/10})$, then (2.11)
follows, and therefore the convergence in distribution of the properly
centered and scaled $LI_{n}$ to the Tracy-Widom distribution also follows.

Beyond the uniform case, in order to evaluate the rate of convergence to the
limiting law for arbitrary distributions, we first need to control how close
$\max_{\begin{subarray}{c}\lambda\in\Lambda_{d}\end{subarray}}Z_{n}(\lambda)$
is to
$\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}_{d}\end{subarray}}Z_{n}(\lambda)$
where again,

$\Lambda^{\prime}_{d}:=\\{\left({j_{1}}/{n},\dots,{j_{m}}/{n}\right):j_{1}+\dots+j_{m}=n\text{
and }p_{i}\neq p_{\max}\implies j_{i}=0\\}.$

######  Lemma 2.3.

Let $n,m\geq 2,\alpha\geq 1,a=6+3\alpha$ and $\Delta=p_{\max}-p_{2nd}$, where
$p_{2nd}$ is the second highest of the $p_{i}$’s. Let $a\log n\leq
2\sqrt{n}\Delta$. Then,

$\mathds{P}\left(\left|\frac{\max_{\begin{subarray}{c}\lambda\in\Lambda_{d}\end{subarray}}Z_{n}(\lambda)}{\sqrt{p_{\max}}}-\frac{\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}_{d}\end{subarray}}Z_{n}(\lambda)}{\sqrt{p_{\max}}}\right|>\frac{(a\log
n)^{2}}{4\Delta\sqrt{np_{\max}}}+\frac{am\log
n}{\sqrt{np_{\max}}}\right)\leq\frac{2m}{n^{\alpha}}.$

###### Proof.

Our analysis of this result rests upon estimating the variations of
$\widetilde{B^{n}_{i}}$. To do so, let $A_{n}$ be the event:

$\forall i\in\\{1,\dots,m\\},\forall
j\in\\{0,\dots,n-1\\},\forall\ell\in\\{1,\dots,n-j\\},\left|\frac{N^{i}_{j+\ell}-N^{i}_{j}-p^{X}_{i}\ell}{\sqrt{n}}\right|\leq\frac{\sigma_{i}\sqrt{\ell}+1}{\sqrt{n}}a\log{n}.$

By Bernstein’s inequality:

$1-\mathds{P}\left(A_{n}\right)\leq
2m\frac{n(n+1)}{2}\exp\left(-\frac{\frac{1}{2}(\sigma_{i}\sqrt{\ell}+1)^{2}(a\log
n)^{2}}{\ell\sigma^{2}_{i}+\frac{1}{3}(\sigma_{i}\sqrt{\ell}+1)a\log{n}}\right)$
(2.13)

and since

$\frac{\frac{1}{2}(\sigma_{i}\sqrt{\ell}+1)^{2}(a\log
n)^{2}}{\ell\sigma^{2}_{i}+\frac{1}{3}(\sigma_{i}\sqrt{\ell}+1)a\log{n}}\geq\frac{\frac{1}{2}(\sigma_{i}\sqrt{\ell}+1)^{2}(a\log
n)^{2}}{\frac{2}{3}(\sigma_{i}\sqrt{\ell}+1)a\log{n}+\frac{1}{3}(\sigma_{i}\sqrt{\ell}+1)a\log{n}}\wedge\frac{\frac{1}{2}(\sigma_{i}\sqrt{\ell}+1)^{2}(a\log
n)^{2}}{\ell\sigma^{2}_{i}+\frac{1}{2}\ell\sigma^{2}_{i}}\geq\frac{a}{3}\log{n},$
(2.14)

it follows that

$1-\mathds{P}\left(A_{n}\right)\leq
2mn^{2-\frac{a}{3}}\leq\frac{2m}{n^{\alpha}}.$

If $A_{n}$ occurs, then for all $\lambda\in\Lambda_{d}$ and
$i\in\\{1,\dots,m\\}$,

$\widetilde{V^{n}_{i}}(\lambda)\leq
a\log{n}\left(\sigma_{i}\sqrt{\lambda_{i}}+\frac{1}{\sqrt{n}}\right).$ (2.15)

Let $A_{n}$ occur. Let $\lambda\in\Lambda_{d}$. Let $s=\sum_{j:p_{j}\neq
p_{\max}}\lambda_{j}$, let $i\in\\{1,\dots,m\\}$ be such that $p_{i}=p_{\max}$
and let $\lambda^{\prime}$ be defined as: $\lambda^{\prime}_{j}=0$ for all $j$
such that $p_{j}\neq p_{\max}$, $\lambda^{\prime}_{i}=\lambda_{i}+s$, and
$\lambda^{\prime}_{j}=\lambda_{j}$ elsewhere. So
$\lambda^{\prime}\in\Lambda^{\prime}_{d}$ and from (2.15),

$Z_{n}(\lambda^{\prime})\geq
Z_{n}(\lambda)+\sqrt{n}s\Delta-\left(\sigma_{i}\sqrt{s}+\frac{1}{\sqrt{n}}\right)a\log
n-\sum_{j:p_{j}\neq
p_{\max}}\left(\sigma_{j}\sqrt{\lambda_{j}}+\frac{1}{\sqrt{n}}\right)a\log n,$

which leads by the Cauchy-Schwarz inequality to

$Z_{n}(\lambda^{\prime})\geq Z_{n}(\lambda)+\sqrt{n}s\Delta-\sqrt{s}a\log
n-\frac{am\log n}{\sqrt{n}}.$

Hence, since $a\log n\leq 2\sqrt{n}\Delta$,

$Z_{n}(\lambda^{\prime})\geq Z_{n}(\lambda)-\frac{(a\log
n)^{2}}{4\Delta\sqrt{n}}-\frac{am\log n}{\sqrt{n}},$

and finally

$0\leq\max_{\begin{subarray}{c}\lambda\in\Lambda_{d}\end{subarray}}Z_{n}(\lambda)-\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}_{d}\end{subarray}}Z_{n}(\lambda)\leq\frac{(a\log
n)^{2}}{4\Delta\sqrt{n}}+\frac{am\log n}{\sqrt{n}}.$ (2.16)

Therefore,

$\mathds{P}\left(\left|\frac{\max_{\begin{subarray}{c}\lambda\in\Lambda_{d}\end{subarray}}Z_{n}(\lambda)}{\sqrt{p_{\max}}}-\frac{\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}_{d}\end{subarray}}Z_{n}(\lambda)}{\sqrt{p_{\max}}}\right|>\frac{(a\log
n)^{2}}{4\Delta\sqrt{np_{\max}}}+\frac{am\log n}{\sqrt{np_{\max}}}\right)\leq
1-\mathds{P}\left(A_{n}\right)\leq\frac{2m}{n^{\alpha}}.$ (2.17)

∎

We can now deduce our theorem, where below $J_{k}$ is defined as in [6,
Theorem 4.1] ($J_{k}$ has same law as $J_{1,k}+\sqrt{(1-kp_{\max})/k}Z$, where
$Z\sim\mathcal{N}(0,1)$ is independent from the other variables)

######  Theorem 2.4.

Let $n,m\geq 2$ and let $k$ be the multiplicity of $p_{\max}$, then,

$\left|\mathds{P}\left(\frac{LI_{n}-np_{\max}}{\sqrt{np_{\max}}}\geq
x\right)-\mathds{P}\left(J_{k}\geq x\right)\right|\leq
2D(k,p_{\max})\frac{(\log
n)^{2}}{\sqrt{np_{\max}}}\left(\frac{21}{\Delta}+C_{2}m\right),$ (2.18)

where $C_{2}$ is a universal constant, where
$D(1,p_{\max}):=1/\sqrt{2\pi(1-p_{\max})}$, and where for $k\geq 2$,
$D(k,p_{\max}):=\min\left\\{\sqrt{{k}/{2\pi(1-kp_{\max})}},k^{3k}(2\pi
e^{2})^{k/2}\sqrt{{e}/{\pi}}\right\\}$.

###### Proof.

We apply Lemma 2.1, and for
$\lambda\in\Lambda^{\prime}:=\\{\lambda\in\Lambda:p_{i}\neq
p_{\max}\implies\lambda_{i}=0\\}$, let $Z^{\prime}_{n}(\lambda)$ be defined as
$Z_{n}(\lambda)$ but with $B^{n}_{i}$ instead of $\widetilde{B^{n}_{i}}$. For
any $\lambda\in\Lambda^{\prime}$,
$\left|Z_{n}(\lambda)-Z^{\prime}_{n}(\lambda)\right|\leq
2m\sup_{\begin{subarray}{c}i\in\\{1,\dots,m\\}\\\ 0\leq t\leq
1\end{subarray}}\left|\sigma_{i}\widetilde{B^{n}_{i}}(t)-\sigma_{i}B^{n}_{i}(t)\right|$,
hence

$\mathds{P}\left(\left|\sup_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}\end{subarray}}Z_{n}(\lambda)-\sup_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}\end{subarray}}Z^{\prime}_{n}(\lambda)\right|\geq\frac{2C\alpha
m(\log n)^{2}}{\sqrt{n}}\right)\leq\frac{2}{n^{\alpha}}.$ (2.19)

For all $\lambda\in\Lambda^{\prime}$,
$Z_{n}(\lambda)=\sum_{i=1}^{m}\widetilde{V^{n}_{i}}(\lambda)$, so
$\sup_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}\end{subarray}}Z_{n}(\lambda)=\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}_{d}\end{subarray}}Z_{n}(\lambda)$.
So

$\mathds{P}\left(\left|\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}_{d}\end{subarray}}Z_{n}(\lambda)-\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}\end{subarray}}Z^{\prime}_{n}(\lambda)\right|\geq\frac{2C\alpha
m(\log n)^{2}}{\sqrt{n}}\right)\leq\frac{2}{n^{\alpha}}.$ (2.20)

For any $x\in\mathbb{R}$, $\alpha\geq 1$, and $a=6+3\alpha$,

$\left|\mathds{P}\left(\frac{\max_{\begin{subarray}{c}\lambda\in\Lambda_{d}\end{subarray}}Z_{n}(\lambda)}{\sqrt{p_{\max}}}\geq
x\right)-\mathds{P}\left(\frac{\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}\end{subarray}}Z^{\prime}_{n}(\lambda)}{\sqrt{p_{\max}}}\geq
x\right)\right|$

$\displaystyle\leq\mathds{P}\left(\left|\frac{\max_{\begin{subarray}{c}\lambda_{d}\in\Lambda\end{subarray}}Z_{n}(\lambda)}{\sqrt{p_{\max}}}-\frac{\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}_{d}\end{subarray}}Z_{n}(\lambda)}{\sqrt{p_{\max}}}\right|>\frac{(a\log
n)^{2}}{4\Delta\sqrt{np_{\max}}}+\frac{am\log n}{\sqrt{np_{\max}}}\right)$
$\displaystyle+\mathds{P}\left(\left|\frac{\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}_{d}\end{subarray}}Z_{n}(\lambda)}{\sqrt{p_{\max}}}-\frac{\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}\end{subarray}}Z^{\prime}_{n}(\lambda)}{\sqrt{p_{\max}}}\right|>\frac{2C\alpha
m(\log n)^{2}}{\sqrt{np_{\max}}}\right)$
$\displaystyle+\mathds{P}\left(\left|\frac{\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}\end{subarray}}Z^{\prime}_{n}(\lambda)}{\sqrt{p_{\max}}}-x\right|\leq\frac{(a\log
n)^{2}}{4\Delta\sqrt{np_{\max}}}+\frac{am\log
n}{\sqrt{np_{\max}}}+\frac{2C\alpha m(\log n)^{2}}{\sqrt{np_{\max}}}\right).$

Now, with the notations of [6, Theorem 4.1], we see that
$\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}\end{subarray}}Z^{\prime}_{n}(\lambda)/\sqrt{p_{\max}}$
has the same law as $J_{k}$. Indeed, one can check that in [6], $\tilde{B}$ is
an $(m-1)$-dimensional Brownian motion with the same covariance as
$\left(\\!\left(\sigma_{r}B^{n}_{r}-\sigma_{r+1}B^{n}_{r+1}\right)\\!/\\!\sqrt{p_{r}+p_{r+1}-(p_{r}-p_{r+1})^{2}}\right)_{1\leq
r\leq m}$, and rewriting $\tilde{B}$ that way gives exactly
${\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}\end{subarray}}Z^{\prime}_{n}(\lambda)}/{\sqrt{p_{\max}}}$.
Assuming $a\log n\leq 2\sqrt{n}\Delta$, from 2.3, [6, Proposition 3.1 (ii)]
and the inequality $a=6+3\alpha\leq 9\alpha$, one gets

$\displaystyle\left|\mathds{P}\left(\frac{LI_{n}-np_{\max}}{\sqrt{np_{\max}}}\geq
x\right)-\mathds{P}\left(J_{k}\geq x\right)\right|$
$\displaystyle\leq\frac{4m}{n^{\alpha}}+2D(k,p_{\max})\frac{(\log
n)^{2}}{\sqrt{np_{\max}}}\left(\frac{21\alpha^{2}}{\Delta}+\left(2C+\frac{9}{\log
2}\right)\alpha m\right),$ (2.21)

where we refer to [6, Proposition 3.1] for a proof that $D(k,p_{\max})$ is a
bound on the infinity norm of the density of the limiting distribution
$\max_{\begin{subarray}{c}\lambda\in\Lambda^{\prime}\end{subarray}}Z^{\prime}_{n}(\lambda)$.
Taking $\alpha=1$ and using $D(k,p_{\max})\geq 1/\sqrt{2\pi}$, one gets, in
particular, (2.18). Now if we no longer assume that $a\log n\leq
2\sqrt{n}\Delta$, the bound (2.21) (and therefore (2.18)) above is still valid
because its right-hand side is then greater than $1$. ∎

##  3 Concluding remarks

  1. (i)

Theorem 2.4 is slightly weaker than Theorem 4.1 in [6], because of $(\log
n)^{2}$ instead of $\log n$ and of the extra term ${21}/{\Delta}$.

The supplementary $\log n$ comes from the different version of the KMT Theorem
(strong embedding) that is used compared to the one in [6] (weak embedding for
each coordinate). Moreover, the proof that they can be built on a same
probability space with the right covariance has been in question. More
precisely, in [6], the authors use the KMT Theorem to get for each
$i\in\\{1,\dots,m\\}$ a probability space with a version of the random walk
$\widetilde{B^{n}_{i}}$ and a Brownian motion approximating it, but it is not
clear how this implies that on some probability space, there are versions of
$\widetilde{B^{n}_{1}},\dots,\widetilde{B^{n}_{m}}$ satisfying the additional
covariance requirement $\widetilde{B^{n}_{1}}+\dots+\widetilde{B^{n}_{m}}=0$.
However, using Kiefer’s version of the KMT approximation, we have a
probability space with $(U_{i})_{1\leq i\leq n}$ uniform on $[0,1]$ and with
$\widetilde{B^{n}_{1}},\dots,\widetilde{B^{n}_{m}}$ well defined on it (and
satisfying the right covariance relation) as shown in the proof of Lemma 2.1.

Note that to the best of our knowledge, it is still an open problem to know
whether or not the extra $\log n$ or $(\log n)^{2}$ factors in the strong
embedding could be improved. Note, nevertheless, the rate $1/\sqrt{n}$ in the
binary uniform case (see [6, Theorem 5.1]).

The extra term ${21}/{\Delta}$ is needed because if one picks
$\Delta=p_{\max}-p_{2nd}$ very small, there is no hope, for any fixed $m$, to
have a sequence $(A^{m}_{n})_{n\geq 2}$ converging to zero such that for any
distribution $p_{1},\dots,p_{m}$, any $n\geq 2$ and $x\in\mathbb{R}$,

$\left|\mathds{P}\left(\frac{LI_{n}-np_{\max}}{\sqrt{np_{\max}}}\geq
x\right)-\mathds{P}\left(J_{k}\geq x\right)\right|\leq A^{m}_{n}.$ (3.1)

Indeed, assume it is the case for, say, $m=2$. Let us apply this bound to

     * •

Case a) $p_{1}=p_{2}=1/2$, and denote
$\left(LI_{n}-np_{\max}\right)/\sqrt{np_{\max}}$ by $Z_{n}$

     * •

Case b) $p_{1}=1/2+1/2^{n},p_{2}=1/2-1/2^{n}$, and denote
$\left(LI_{n}-np_{\max}\right)/\sqrt{np_{\max}}$ by $Z^{\prime}_{n}$.

We get

$\mathds{P}\left(Z_{n}\geq x\right)-\mathds{P}\left(J_{2}\geq
x\right)-\mathds{P}\left(Z^{\prime}_{n}\geq x\right)+\mathds{P}\left(J_{1}\geq
x\right)\xrightarrow[n\to\infty]{}0.$

There is a coupling such that with high probability, the letters
$X_{1},\dots,Y_{n}$ in case a) are all equal to the letters in case b), hence

$\mathds{P}\left(Z_{n}\geq x\right)-\mathds{P}\left(Z^{\prime}_{n}\geq
x\right)\xrightarrow[n\to\infty]{}0.$

Furthermore, $J_{1}$ has distribution
$\mathcal{N}\left(0,\frac{1}{4}-\frac{1}{2^{n+1}}\right)$ so

$\mathds{P}\left(J_{1}\geq
x\right)\xrightarrow[n\to\infty]{}\mathds{P}\left(Z_{1/4}\geq x\right),$

where $Z_{1/4}\sim\mathcal{N}\left(0,1/4\right)$. Putting together these three
limits, we get

$\mathds{P}\left(J_{2}\geq x\right)=\mathds{P}\left(Z_{1/4}\geq x\right),$

which means that the limiting law for the uniform binary case is normal.
However, this is known to be false. So, $(A^{m}_{n})_{n\geq 2}$ has to depend
on the distribution. We can actually find a contradiction as soon as
$\Delta=o\left(1/\sqrt{n}\right)$. Our bound (2.18), on the other hand, is not
exposed to this kind of cases because if the right-hand side is less than $1$
then $\Delta\geq{\log n}/{\sqrt{n}}$. So Theorem 4.1 in [6] only holds for $n$
large enough, and not for all $n\geq 2$.

  2. (ii)

As before, let us no longer consider $m,k$ and the distribution
$p_{1},\dots,p_{m}$ to be fixed. We assume that both $m$ and $k$ converge to
infinity with $n$, and aim to find for which sequences we have for all
$\varepsilon>0$,

$\mathds{P}\left(\left|\left(\frac{LI_{n}-np_{\max}}{\sqrt{np_{\max}}}-2\sqrt{k}\right)k^{1/6}-\left(J_{k}-2\sqrt{k}\right)k^{1/6}\right|\geq\varepsilon\right)\xrightarrow[n\to\infty]{}0.$
(3.2)

Then, since $\left(J_{k}-2\sqrt{k}\right)k^{1/6}$ has the same distribution as
$\left(J_{1,k}-2\sqrt{k}\right)k^{1/6}+k^{1/6}\sqrt{\left(1-kp_{\max}\right)/k}Z$,
it will converge in distribution to $F_{TW}$, implying that
$\left(\left(LI_{n}-np_{\max}\right)/\sqrt{np_{\max}}-2\sqrt{k}\right)k^{1/6}$
converges to $F_{TW}$ as well.

Applying the Lemma 2.3 and the bound (2.19) lead to

$\mathds{P}\left(\left|\left(\frac{LI_{n}-np_{\max}}{\sqrt{np_{\max}}}-2\sqrt{k}\right)k^{1/6}-\left(J_{k}-2\sqrt{k}\right)k^{1/6}\right|\geq\frac{(\log
n)^{2}}{\sqrt{np_{\max}}}\left(\frac{21}{\Delta}+C_{2}m\right)\right)\leq\frac{4m}{n^{\alpha}}.$
(3.3)

Taking $m=o((n/(\log n)^{4})^{3/10})$ (as previously done in the uniform case)
and $m=o(\sqrt{np_{\max}}\Delta/(\log n)^{2})$, then (3.2) follows. Note that
in particular, when the conditions of Theorem 6 in [1] are satisfied, the
condition $m=o(\sqrt{np_{\max}}\Delta/\log n)$ follows. This is not enough to
conclude, first, we need $(\log n)^{2}$ instead of $\log n$, and more
importantly, the first condition $m=o((n/(\log n)^{4})^{3/10})$ is missing.
This is an omission in [1], because as it is, there is no condition on $m$ and
this leads to a counterexample. Indeed, let
$k=n^{1/9},p_{\max}=n^{-2/3},m=2^{n}+k,p_{2nd}=\left(1-n^{5/9}\right)/2^{n}$,
it is easy to check that the conditions of Theorem 6 there, hold true but its
conclusion does not. The two conditions on $m$ we give above do fix this
issue.

  3. (iii)

To finish these notes, let us investigate the convergence of moments which, in
particular, will provide a speed of convergence result in the distance
$W_{p}$, $p\geq 1$, given by

$W_{p}(X,Y):=\inf_{\begin{subarray}{c}X^{\prime}\text{ has same law as }X\\\
Y^{\prime}\text{ has same law as
}Y\end{subarray}}\left(\mathds{E}\left|X^{\prime}-Y^{\prime}\right|^{p}\right)^{1/p}.$

Let us start with the uniform case. We have seen that

$\mathds{P}\left(\left|T_{k,m}-J_{k,m}\right|\geq
2k(m-k+1)\sqrt{m}C\frac{\alpha(\log
n)^{2}}{\sqrt{n}}\right)\leq\frac{2}{n^{\alpha}}.$ (3.4)

In particular, taking $\alpha=p$, and setting
$\varepsilon_{n,m,k}:=2k(m-k+1)\sqrt{m}C{p(\log n)^{2}}/{\sqrt{n}}$, we have

$\displaystyle\mathds{E}\left|T_{k,m}-J_{k,m}\right|^{p}$
$\displaystyle\leq\mathds{E}\left(\left|T_{k,m}-J_{k,m}\right|^{p}\mathds{1}_{\left|T_{k,m}-J_{k,m}\right|\leq\varepsilon(n,m,k)}+\left|T_{k,m}-J_{k,m}\right|^{p}\mathds{1}_{\varepsilon(n,m,k)<\left|T_{k,m}-J_{k,m}\right|<2m\sqrt{n}}\right.$
$\displaystyle\hskip
150.69397pt\left.+\left|T_{k,m}-J_{k,m}\right|^{p}\mathds{1}_{2m\sqrt{n}\leq\left|T_{k,m}-J_{k,m}\right|}\right)$
$\displaystyle\leq\varepsilon(n,m,k)^{p}+(2m\sqrt{n})^{p}\frac{2}{n^{p}}+\mathds{E}\left|2J_{k,m}\right|^{p}\mathds{1}_{m\sqrt{n}\leq\left|J_{k,m}\right|},$

(since $\left|T_{k,m}\right|\leq\sqrt{n}\leq m\sqrt{n}$). Using, once more,
(2.4) and an integration by parts, one gets for some constant $C(p)$

$\mathds{E}\left|2J_{k,m}\right|^{p}\mathds{1}_{m\sqrt{n}\leq\left|J_{k,m}\right|}\leq
C(p)\left(\frac{m}{\sqrt{n}}\right)^{p},$ (3.5)

and therefore, for some constant $C^{\prime}(p)$,

$W_{p}(T_{k,m},J_{k,m})\leq\mathds{E}\left|T_{k,m}-J_{k,m}\right|^{p}\leq
C^{\prime}(p)\varepsilon(m,n,k).$ (3.6)

  4. (iv)

Theorem 2.4 can be generalized to longest common and increasing subsequences.
We use the notations and framework in [3]. Additionally, let
$Z_{n}={\left(LCI_{n}-ne_{\max}\right)}/{\sqrt{n}}$ and let $Z$ be its
limiting distribution, that is, either $Z^{a}$ or $Z^{b}$ depending on the
distributions $p^{X},p^{Y}$. The proof of Lemma 2.2 in [3], taking
$\eta=1/12$, holds for the sequence
$(\varepsilon_{n})=C_{1}(m,p^{X},p^{Y})n^{-1/8}$, where $C_{1}(m,p^{X},p^{Y})$
is a constant depending on $m,p^{X},p^{Y}$. To study the cases a) and b) under
the same umbrella, let us define the function
$L:\left(\mathbb{R}^{m}\right)^{2}\rightarrow\mathbb{R}$ by: for all
$(v^{X},v^{Y})\in\left(\mathbb{R}^{m}\right)^{2}$, in case a),
$L(v^{X},v^{Y})=\sum_{i=1}^{m}v^{X}$, and in case b),
$L(v^{X},v^{Y})=\mathfrak{m}(v^{X},v^{Y})$. It is not hard to see, from the
expression of $\mathfrak{m}$ in Lemma 1.5 in [3], that in each case, there is
a constant $C_{2}(m,p^{X},p^{Y})$ such that $L$ is
$C_{2}(m,p^{X},p^{Y})$-Lipschitz for the $\ell^{1}$-distance on
$\left(\mathbb{R}^{m}\right)^{2}$. Now, as a consequence of Lemma 2.2 there,
it follows that if $B^{n,X},B^{n,Y}$ are in $E_{n}^{1/12}$, then

$\left|Z_{n}-\max_{\lambda\in
U}L(V^{n,X},V^{n,Y})\right|\leq\frac{C_{1}(m,p^{X},p^{Y})}{n^{1/8}},$ (3.7)

where $U=J$ in case a) and $U=K_{\Lambda^{2}}$ in case b). Therefore,

$\mathds{P}\left(\left|Z_{n}-\max_{\lambda\in
U}L(V^{n,X},V^{n,Y})\right|>\frac{C_{1}(m,p^{X},p^{Y})}{n^{1/8}}\right)\leq
1-\mathds{P}\left(A^{1/12}_{n}\right)\leq\frac{C(m)}{n},$ (3.8)

where $C(m)$ is a constant (recalling (1.9) in [3]). Furthermore, from the
expression of the limiting distribution in Theorem 2.1, we see that in any
case a) or b), the limiting distribution may be written as $L(B^{X},B^{Y})$.
We construct, with our Lemma 2.1, two Brownian motions
$\widehat{B^{X}_{n}},\widehat{B^{Y}_{n}}$ "close" to $B^{n,X},B^{n,Y}$, on the
same probability space (this is possible by applying the lemma twice, and then
taking the product space). Let $\widehat{Z_{n}}=\max_{\lambda\in
U}L(\widehat{V^{n,X}},\widehat{V^{n,Y})}$, it has same distribution as the
limiting distribution $Z$, and

$\mathds{P}\left(\left|\max_{\lambda\in
U}L(V^{n,X},V^{n,Y})-\widehat{Z_{n}}\right|>2mC_{2}(m,p^{X},p^{Y})\varepsilon\right)\leq\mathds{P}\left(\sup_{\begin{subarray}{c}i\in\\{1,\dots,m\\}\\\
0\leq t\leq
1\end{subarray}}\left|\sigma_{i}\widehat{B^{n,X}_{i}}(t)-\sigma_{i}B^{n,X}_{i}(t)\right|\geq\varepsilon\right)\\\
+\mathds{P}\left(\sup_{\begin{subarray}{c}i\in\\{1,\dots,m\\}\\\ 0\leq t\leq
1\end{subarray}}\left|\sigma_{i}\widehat{B^{n,Y}_{i}}(t)-\sigma_{i}B^{n,Y}_{i}(t)\right|\geq\varepsilon\right),$
(3.9)

so in particular, applying Lemma 2.1,

$\mathds{P}\left(\left|\max_{\lambda\in
U}L(V^{n,X},V^{n,Y})-\widehat{Z_{n}}\right|>2mC_{2}(m,p^{X},p^{Y})C\frac{\alpha(\log
n)^{2}}{\sqrt{n}}\right)\leq\frac{4}{n^{\alpha}}.$ (3.10)

Putting together (3.8) and (3.10), with $\alpha=1$, gives

$\mathds{P}\left(\left|Z_{n}-\widehat{Z_{n}}\right|>\frac{C_{1}(m,p^{X},p^{Y})}{n^{1/8}}+2mC_{2}(m,p^{X},p^{Y})C\frac{(\log
n)^{2}}{\sqrt{n}}\right)\leq\frac{C(m)+4}{n}.$ (3.11)

If, just as in the single subsequence case, a bound $D(m,p^{X},p^{Y})$ on the
density of $Z$ is possible, then we can conclude in the same way: For all
$x\in\mathbb{R}$,

$\displaystyle\left|\mathds{P}\left(Z_{n}\geq x\right)-\mathds{P}\left(Z\geq
x\right)\right|$
$\displaystyle\leq\mathds{P}\left(\left|Z_{n}-\widehat{Z_{n}}\right|>\frac{C_{1}(m,p^{X},p^{Y})}{n^{1/8}}+2mC_{2}(m,p^{X},p^{Y})C\frac{(\log
n)^{2}}{\sqrt{n}}\right)$
$\displaystyle+\mathds{P}\left(\left|\widehat{Z_{n}}-x\right|\leq\frac{C_{1}(m,p^{X},p^{Y})}{n^{1/8}}+2mC_{2}(m,p^{X},p^{Y})C\frac{(\log
n)^{2}}{\sqrt{n}}\right)$
$\displaystyle\leq\frac{C(m)+4}{n}+D(m,p^{X},p^{Y})\left(\frac{C_{1}(m,p^{X},p^{Y})}{n^{1/8}}+2mC_{2}(m,p^{X},p^{Y})C\frac{(\log
n)^{2}}{\sqrt{n}}\right)$
$\displaystyle\leq\frac{C_{3}(m,p^{X},p^{Y})}{n^{1/8}}.$

Note that the exponent $-1/8$ can be improved taking a smaller $\eta$, but
only up to $-1/4$. This is because in [3], the focus was to get convergence in
distribution, rather than a tight bound. It might even be possible to get
$(\log n)^{2}/\sqrt{n}$ instead.

Acknowledgements: Many thanks to Jon Wellner for his bibliographical help on
KMT approximations, to Ryan O’Donnell and Boris Bukh for their inquiries on
previous results, and to the ICTS in Bengaluru as well as the GESDA program at
IHP in Paris for their hospitality and support while part of this research was
carried out.

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# Hartree–Fock Critical Nuclear Charge in Two-Electron Atoms

Hugh G. A. Burton<EMAIL_ADDRESS>Physical and Theoretical
Chemistry Laboratory, Department of Chemistry, University of Oxford, South
Parks Road, Oxford, OX1 3QZ, U.K.

###### Abstract

Electron correlation effects play a key role in stabilising two-electron atoms
near the critical nuclear charge, representing the smallest charge required to
bind two electrons. However, deciphering the importance of these effects
relies on fully understanding the uncorrelated Hartree–Fock description. Here,
we investigate the properties of the ground state wave function in the small
nuclear charge limit using various symmetry-restricted Hartree–Fock
formalisms. We identify the nuclear charge where spin-symmetry breaking occurs
to give an unrestricted wave function that predicts the ionisation of an
electron. We also discover another critical nuclear charge where the closed-
shell electron density detaches from the nucleus, and identify the importance
of fractional spin errors and static correlation in this limit.

##  I Introduction

How much positive charge is required to bind two electrons to a nucleus? This
simple question has been subject to intense research and debate ever since the
early 1930s. Hylleraas (1930); Stillinger (1966); Stillinger and Weber (1974);
Ivanov (1995a); Baker  _et al._ (1990); Armour, Richard, and Varga (2005);
Estienne  _et al._ (2014); King  _et al._ (2015); Olivares Pilón and Turbiner
(2015) High-precision calculations have only recently converged on a critical
nuclear charge for binding two electrons of
$Z_{\text{c}}~{}=~{}0.911\,028\,224\,077\,255\,73(4)$.Estienne  _et al._
(2014); King  _et al._ (2015); Olivares Pilón and Turbiner (2015) For
$Z>Z_{\text{c}}$, the two-electron atom ($Z\,\text{e}\,\text{e}$) is bound and
stable, with an energy lower than the ionised system ($Z\,\text{e}+\text{e}$).
However, for $Z<Z_{\text{c}}$, the energy of the bound atom becomes higher
than the ionised system, causing an electron to spontaneously detach from the
nucleus. As a result, the critical charge marks the threshold for stability in
the three-body problemKais and Shi (2000); Armour, Richard, and Varga (2005);
King, Longford, and Cox (2013) and can be interpreted as a quantum phase
transition.Kais, Wegner, and Wei (2006)

While the critical nuclear charge is fascinating in its own right, the two-
electron atom also provides an essential model for understanding the
performance of electronic structure approximations. It is the simplest
chemical system where electron correlation is present, which is thought to be
essential in binding the two electrons near $Z_{\text{c}}$.Stillinger and
Weber (1974); King, Baskerville, and Cox (2018); Baskerville, King, and Cox
(2019) In particular, comparing the closed-shell HF energy to the exact energy
of the ionised system shows that HF theory fails to predict a stable two-
electron atom with $Z_{\text{c}}<Z<1.031\,177\,528$,King, Baskerville, and Cox
(2018) including H-.Goddard (1968); Cox  _et al._ (2020) However, interpreting
exactly _how_ correlation influences the stability of the two-electron atom is
made difficult by an incomplete understanding of the HF approximation for
small $Z$.

In many ways, placing artificial restrictions on the wave function makes the
HF description more complicated to interpret than the exact result. For
example, the restricted HF (RHF) formalism can only predict doubly-occupied
orbitals,Szabo and Ostlund (1989) and one might ask if comparing the RHF
energy to the exact one-electron energy is a fair way to identify the RHF
critical charge. Alternatively, the unrestricted HF (UHF) approach allows the
spin-up and spin-down electrons to occupy different spatial orbitals,Szabo and
Ostlund (1989) providing a qualitatively correct model for the dissociation of
a single bound electron in H- at the expense of broken spin-symmetry.Cox  _et
al._ (2020); Goddard (1968) The onset of HF symmetry breaking is marked by
instability thresholds in the orbital Hessian,Seeger and Pople (1977) which
have also been interpreted as critical charges in closed-shell atoms.Uhlířová
and Zamastil (2020) These sudden qualitative changes in the HF wave function
can also be probed using the average radial electronic positions, providing an
alternative indicator for electron ionisation that does not rely on energetic
comparisons with the exact result. However, to the best of our knowledge, the
exact nuclear charge for UHF symmetry breaking, and the qualitative properties
of HF wave functions near this point, remain unknown.

Previous studies on the two-electron atom using HF theory have primarily
focussed on the large $Z$, or “high-density”, limit (see e.g., Ref. Loos and
Gill, 2010a). In this limit, the closed-shell RHF wave function provides a
good approximation to the exact result, creating a model for understanding
dynamic correlation.Loos and Gill (2009, 2010a) Alternatively, the small-$Z$
“low-density” limit, where static correlation becomes significant, remains far
less explored. The primary challenges of small $Z$ include the presence of HF
symmetry-breaking and convergence issues that occur with diffuse basis
functions. One recent HF study has been unable to reliably converge the RHF
approximation for $Z<0.85$,King, Baskerville, and Cox (2018) hindering
attempts to understand HF theory for smaller $Z$. Consequently, the small-$Z$
limit also provides a model for understanding how to predict strong static
correlation,Hollett and Gill (2011) which remains a major computational
challenge.

In this contribution, we investigate the properties of the RHF and UHF ground-
state wave functions in the small $Z$ limit. We use numerical Laguerre-based
HF calculations to compute the exact location of the UHF symmetry-breaking
threshold. By investigating the average radial positions in the RHF and UHF
wave functions, we assess how each HF formalism predicts electron detachment
near the critical charge. Our results suggest that the UHF symmetry-breaking
threshold represents the onset of ionisation and forms a branch point
singularity in the complex $Z$ plane. Alternatively, RHF theory predicts a
closed-shell critical point where half the electron density becomes ionised,
leading to strong static correlation for small $Z$.

##  II Two-Electron Atomic Hamiltonian

The $Z$-scaled Hamiltonian for the two-electron atom with an infinite nuclear
mass isHylleraas (1930)

$\mathcal{H}=-\frac{1}{2}\quantity(\boldsymbol{\nabla}_{1}^{2}+\boldsymbol{\nabla}_{2}^{2})-\frac{1}{\rho_{1}}-\frac{1}{\rho_{2}}+\frac{1}{Z}\frac{1}{\rho_{12}},$
(1)

where $\rho_{i}=r_{i}/Z$ is the scaled distance of electron $i$ from the
nucleus, $\rho_{12}=|\bm{\rho}_{1}-\bm{\rho}_{2}|$ is the scaled inter-
electronic distance, and the unscaled distances have atomic units $a_{0}$.
Nuclear charges are given in atomic units $e$. The exact wave function is
defined by the time-independent Schrödinger equation

$\mathcal{H}\Psi(\bm{x}_{1},\bm{x}_{2})=\tilde{E}\Psi(\bm{x}_{1},\bm{x}_{2})$
(2)

with the spin-spatial coordinate $\bm{x}_{i}=(\bm{\rho}_{i},\sigma_{i})$ and
the scaled energy $\tilde{E}=E/Z^{2}$. The electron-electron repulsion can be
considered as a perturbation to the independent-particle model with the
coupling strength $\lambda=1/Z$,Hylleraas (1930) giving the power series
expansions $\tilde{E}(\lambda)=\sum_{k=0}^{\infty}\tilde{E}^{(k)}\lambda^{k}$
and $\Psi(\lambda)=\sum_{k=0}^{\infty}\Psi^{(k)}\lambda^{k}$. The critical
nuclear charge $Z_{\text{c}}$ can then be identified from the radius of
convergence of these series,Stillinger (1966); Baker  _et al._ (1990); Ivanov
(1995a, b) defined by the distance of the closest singularity to the origin in
the complex $\lambda$ plane.Marie, Burton, and Loos (2020) Both $E(\lambda)$
and $|\Psi(\lambda)|^{2}$ have complicated singularities on the positive real
axis at $\lambda_{\text{c}}=1/Z_{\text{c}}$,Baker  _et al._ (1990) which have
been interpreted as a quantum phase transition in the complete-basis-set
limit.Kais, Wegner, and Wei (2006)

The HF wave function is a single Slater determinant
$\Psi_{\text{HF}}(\bm{x}_{1},\bm{x}_{2})$ built from the antisymmetrised
product of the occupied spin-orbitals $\psi_{i}(\bm{x})$. These orbitals are
self-consistent eigenfunctions of the one-electron Fock operator
$\hat{f}(\bm{x})$, with the corresponding eigenvalues defining orbital
energies. The $Z$-scaled Fock operator is

$\hat{f}(\bm{x})=\hat{h}(\bm{x})+\frac{1}{Z}\sum_{i=1}^{2}\quantity[\hat{J}_{i}(\bm{x})-\hat{K}_{i}(\bm{x})],$
(3)

with the one-electron Hamiltonian

$\hat{h}(\bm{x})=-\frac{\boldsymbol{\nabla}^{2}}{2}-\frac{1}{\rho},$ (4)

and the Coulomb and exchange operators denoted as $\hat{J}_{i}(\bm{x})$ and
$\hat{K}_{i}(\bm{x})$ respectively (see Ref. Szabo and Ostlund, 1989). The
total HF energy is

$\tilde{E}^{\text{HF}}=\frac{1}{2}\sum_{i=1}^{2}(h_{i}+f_{i}),$ (5)

with the matrix elements $h_{i}=\langle\psi_{i}|\hat{h}|\psi_{i}\rangle$ and
$f_{i}=\langle\psi_{i}|\hat{f}|\psi_{i}\rangle$.

The self-consistent two-electron component of the Fock operator can be
considered as a perturbation with the coupling strength $\lambda=1/Z$. For
large $Z$ ($\lambda\to 0$), only the one-electron component remains and the HF
wave function is exact.Loos and Gill (2010a) As $Z$ becomes smaller and
$\lambda$ grows, the self-consistent repulsion becomes increasingly dominant
over the one-electron nuclear attraction. Eventually, it becomes energetically
favourable for a pair of lower-energy UHF solutions to emerge where either the
spin-up or spin-down electron becomes detached from the nucleus.Goddard
(1968); Cox  _et al._ (2020) This phenomenon is analogous to the
Coulson–Fischer point in stretched H2, where the spin-up and spin-down
electrons localise on opposite atoms,Coulson and Fischer (1949) and is closely
related to Wigner crystallisation.Wigner (1934) By analytically continuing an
equivalent two-electron coupling parameter to complex values, we have recently
shown that the UHF wave functions form a non-Hermitian square-root branch
point at the symmetry-breaking threshold.Burton, Thom, and Loos (2019); Marie,
Burton, and Loos (2020) Remarkably, following a complex-valued contour around
this point leads to the interconversion of the degenerate solutions, and
allows a ground-state wave function to be smoothly evolved into an excited-
state wave function.Burton, Thom, and Loos (2019)

##  III Computational Details

In the present work, we follow Ref. King, Baskerville, and Cox, 2018 and
express the spatial component $\phi_{p}(\bm{r})$ of the HF spin-orbital
$\psi_{p}(\bm{x})$ using the spherically-symmetric Laguerre-based
functionsRiley, Hobson, and Bence (2006)

$\chi_{\mu}(\bm{r})=\exp(-\frac{Ar}{2})\ L^{(1)}_{\mu}(Ar),$ (6)

giving

$\phi_{p}(\bm{r})=\sum_{\mu=0}^{\infty}\chi_{\mu}\quantity(\bm{r})C^{\mu\cdot}_{\cdot
p}.$ (7)

Here we employ the nonorthogonal tensor notation of Head–Gordon et al.Head-
Gordon, Maslen, and White (1998) The non-linear parameter $A$ controls the
spatial extent of the basis functions and is optimised alongside the
coefficients $C^{\mu\cdot}_{\cdot p}$.King, Baskerville, and Cox (2018) In
practice, this expansion is truncated at a finite basis set of size $n$. To
avoid previous issues with iterative solutions to the HF equations for small
$Z$, we optimise the $C^{\mu\cdot}_{\cdot p}$ coefficients for a fixed $A$
value using the quasi-Newton Geometric Direct Minimisation (GDM) algorithm.Van
Voorhis and Head-Gordon (2002) The optimal $A$ value is then identified
through another quasi-Newton minimisation with the orbital coefficients re-
optimised on each step. All calculations were performed in a developmental
version of Q-Chem,Y. Shao, Z. Gan, E. Epifanovsky, A. T. Gilbert, M. Wormit,
J. Kussmann, A. W. Lange, A. Behn, J. Deng, X. Feng, et al. (2015) and
analytic expressions for the Laguerre-based integrals are provided in the
Supporting Information.

##  IV Results

###  IV.1 Spin-Symmetry Breaking Critical Point

First, we identify the critical nuclear charge for HF symmetry-breaking
$Z_{\text{c}}^{\text{UHF}}$ using a bisection method to locate the point where
the lowest orbital Hessian eigenvalue of the RHF solution vanishes.Seeger and
Pople (1977) The convergence of $Z_{\text{c}}^{\text{UHF}}$ with respect to
the basis set size is shown in Table 1, giving a best estimate of
$Z_{\text{c}}^{\text{UHF}}=1.057\,660\,253\,46(1)$. This value is converged
for $n\geq 24$, for which converged RHF energies for He and H- are also
obtained as

$\displaystyle E_{\text{RHF}}(\text{He})$
$\displaystyle=-2.861\,679\,995\,612\,23(1)$ $\displaystyle
E_{\text{RHF}}(\text{\ce{H^{-}}})$
$\displaystyle=-0.487\,929\,734\,370\,84(1),$

in agreement with the best variational benchmarks up to 10 decimal
places.King, Baskerville, and Cox (2018); Lehtola (2019); Roothaan and Soukup
(1979) We believe that this is the first numerically precise estimate of a
symmetry-breaking threshold in the complete-basis-set HF limit, and therefore
defines a new type of benchmark value within electronic structure theory. As
expected, we find $Z_{\text{c}}^{\text{UHF}}>1$, and thus our result is
consistent with previous observations of UHF symmetry breaking in the hydride
anion.Cox  _et al._ (2020); Goddard (1968)

Table 1: Convergence of the UHF symmetry-breaking threshold $Z_{\text{c}}^{\text{UHF}}$ and the associated energy $E^{\text{UHF}}(Z_{\text{c}}^{\text{UHF}})$ with respect to basis set size. Best estimates and quoted errors correspond to the mean and standard deviation of the converged values $n\geq 24$ respectively. $n$ | $Z_{\text{c}}^{\text{UHF}}/e$ | $E^{\text{UHF}}(Z_{\text{c}}^{\text{UHF}})/\text{E}_{\text{h}}$
---|---|---
10 | $1.057\,651\,800\,057$ | $-0.570\,335\,516\,87$
12 | $1.057\,658\,412\,462$ | $-0.570\,345\,373\,24$
14 | $1.057\,659\,966\,054$ | $-0.570\,347\,687\,12$
16 | $1.057\,660\,213\,291$ | $-0.570\,348\,055\,22$
18 | $1.057\,660\,248\,206$ | $-0.570\,348\,107\,19$
20 | $1.057\,660\,252\,818$ | $-0.570\,348\,114\,05$
22 | $1.057\,660\,253\,391$ | $-0.570\,348\,114\,91$
24 | $1.057\,660\,253\,461$ | $-0.570\,348\,115\,01$
26 | $1.057\,660\,253\,464$ | $-0.570\,348\,115\,01$
28 | $1.057\,660\,253\,462$ | $-0.570\,348\,115\,01$
30 | $1.057\,660\,253\,464$ | $-0.570\,348\,115\,02$
32 | $1.057\,660\,253\,439$ | $-0.570\,348\,114\,98$
34 | $1.057\,660\,253\,473$ | $-0.570\,348\,115\,03$
36 | $1.057\,660\,253\,458$ | $-0.570\,348\,115\,01$
38 | $1.057\,660\,253\,477$ | $-0.570\,348\,115\,03$
40 | $1.057\,660\,253\,466$ | $-0.570\,348\,115\,02$
Best | $1.057\,660\,253\,46(1)$ | $-0.570\,348\,115\,01(2)$
Figure 1: $Z$-scaled energy (top) and average radial position $\langle
r\rangle$ (middle) using various HF formalisms and exact one- and two-electron
results. Exact two-electron data are reproduced from Ref. King  _et al._ ,
2015. The lowest two Hessian eigenvalues for the RHF solution (bottom) show
the onset of UHF symmetry breaking and a persistent zero eigenvalue for small
$Z$.

Figure 1 (top panel) compares the $Z$-scaled RHF energy (red) and the
symmetry-broken UHF energy (blue dashed) as functions of $Z^{-1}$ with $n=26$.
We also consider the exact one-electron energy (black) that corresponds to the
ionised atom, the exact two-electron energy (grey dashed; reproduced from Ref.
King  _et al._ , 2015), and a fractional spin RHF calculation (dashed orange)
with half a spin-up and half spin-down electron (see Sec. IV.3). The UHF
symmetry-breaking threshold $Z_{\text{c}}^{\text{UHF}}$ (black dot) occurs
below the exact one-electron energy, and thus $Z_{\text{c}}^{\text{UHF}}$ is
greater than the HF critical nuclear charge previously identified using
energetic arguments.King, Baskerville, and Cox (2018) This suggests that the
RHF approximation is already an inadequate representation of the exact wave
function before it becomes degenerate with the one-electron atom. Beyond this
point, the RHF energy continues to increase, while the UHF energy rapidly
flattens towards the exact one-electron result. There is therefore a small
relaxation region during which the UHF approximation approaches a qualitative
representation of the one-electron atom.

Radial electron position expectation values $\langle r\rangle$ provide further
insights into the properties of the two-electron atom close to electron
detachment.King  _et al._ (2015); Baskerville, King, and Cox (2019); King,
Rhodes, and Cox (2018) The exact wave function yields an “inner” and “outer”
electron, with repulsive interactions pushing the inner electron closer to the
nucleus than in the corresponding hydrogenic system.King, Rhodes, and Cox
(2018) For $Z>Z_{\text{c}}^{\text{UHF}}$, the RHF radial electron position
closely matches the averaged exact two-electron result (grey dashed). However,
the RHF result starts to deviate from the two-electron value as electron
correlation effects become significant for $Z<Z_{\text{c}}^{\text{UHF}}$. In
contrast, the additional flexibility of the UHF wave function correctly
predicts the separation of an inner and outer electron. This ionisation occurs
almost immediately for $Z<Z_{\text{c}}^{\text{UHF}}$, as indicated by a sudden
increase in $\langle r\rangle$ for the dissociating electron (Fig. 1: middle
panel), while the bound electron tends towards the exact one-electron result.
Comparing the radial distribution functions $P(r)=r^{2}|\psi(r)|^{2}$ for each
electronic orbital at $Z=1$ (Fig. 2), we find that the inner UHF orbital
closely matches the one-electron H atom, which is also the case for the exact
wave function.King, Rhodes, and Cox (2018) However, at $Z=1$, the outer
electron has essentially ionised from the atom in the UHF approximation, but
remains closely bound to the nucleus in the fully-correlated description. The
UHF wave function therefore rapidly approximates the ionised atom for
$Z<Z_{\text{c}}^{\text{UHF}}$, and $Z_{\text{c}}^{\text{UHF}}$ can be
interpreted as a critical charge for a stable two-electron atom. This
approximation essentially overlocalises the electron density between
$Z_{\text{c}}<Z<Z_{\text{c}}^{\text{UHF}}$ (including H-), as previously
observed for two-electrons on concentric spheres,Loos and Gill (2010b) and
fails to capture the correlation required to describe the exact critical
charge.

Figure 2: Radial distribution functions for different HF orbitals compared to
the exact one-electron wave function at $Z=1$.

Figure 1 also indicates that the dissociation of the outer electron for the
UHF approximation is relatively sudden and closely mirrors the behaviour at
the exact critical nuclear charge.King  _et al._ (2015) Furthermore, there is
small region where the average radial position of the inner electron tends
towards the one-electron result. It is known that the exact two-electron
system exhibits a shape resonance as the nuclear charge goes through the
critical point, with the outer electron remaining at a finite distance from
the nucleus.Estienne  _et al._ (2014); Hoffmann-Ostenhof, Hoffman-Ostenhof,
and Simon (1983) One might therefore interpret the region where the UHF
electron positions tend towards the one-electron result as an approximation of
this resonant stability regime.

For $Z<Z_{\text{c}}$, the exact wave function is an equal combination of two
configurations where either the spin-up or spin-down electron remains bound to
the nucleus. In contrast, the single-determinant nature of the UHF wave
function means that only one of these configurations can be represented: the
UHF orbitals are “pinned” to one resonance form.Trail, Towler, and Needs
(2003) There must therefore be a wave function singularity at
$Z_{\text{c}}^{\text{UHF}}$ where the UHF approximation branches into a form
with either the spin-up or spin-down electron remaining bound. The
mathematical structure of this point can be revealed by following a continuous
pathway around $Z_{\text{c}}^{\text{UHF}}$ in the complex $Z$ plane. When $Z$
is analytically continued to complex values, the Fock operator becomes non-
Hermitian and we must consider the holomorphic HF approach.Hiscock and Thom
(2014); Burton and Thom (2016); Burton, Gross, and Thom (2018) In the
remainder of this Section, we fix the non-linear $A$ parameter to its value at
$Z_{\text{c}}^{\text{UHF}}$ as the non-Hermitian energy is complex-valued and
cannot be variationally optimised.

Figure 3 shows the real component of $\langle r\rangle$ for the (initial)
inner electron along a pathway which spirals in towards
$Z_{\text{c}}^{\text{UHF}}$, parametrised as

$Z(\xi)=Z_{\text{c}}^{\text{UHF}}-\quantity(0.02-\frac{0.001\xi}{2\pi})\exp(\text{i}\xi).$
(8)

Remarkably, after one complete rotation ($\xi=2\pi$), the inner and outer
electrons have swapped, indicating that the degenerate UHF solutions have been
interconverted. A second full rotation is required to return the states to
their original forms. The two degenerate UHF wave functions are therefore
connected as a square-root branch point in the complex-$Z$ plane, in agreement
with our previous observations in analytically solvable models.Burton, Thom,
and Loos (2019); Marie, Burton, and Loos (2020) Furthermore, the branch point
behaves as a quasi-exceptional point, where the two solutions become identical
but remain normalised (see Ref. Burton, Thom, and Loos, 2019), providing the
first example of this type of non-Hermitian HF degeneracy in the complete-
basis-set limit.

Figure 3: Average radial position $\langle r\rangle$ of the inner electron
along a spiral contour in the complex $Z$ plane converging on
$Z_{\text{c}}^{\text{UHF}}$ using $n=26$. On each rotation, the UHF wave
function transitions between the two degenerate solutions.

###  IV.2 Closed-Shell Critical Point

We now consider the fate of the RHF ground state as $Z$ continues to decrease
below $Z_{\text{c}}^{\text{UHF}}$. Intuitively, one might expect that the
doubly-occupied RHF orbitals would be unable to describe the open-shell atom
with an ionised electron. Indeed, King et al. have observed a smooth and
finite $\langle r\rangle$ value for the RHF wave function as low as $Z=0.85$,
with erratic convergence for lower nuclear charges.King, Baskerville, and Cox
(2018) A similar nuclear charge $Z=0.84$ was identified in Ref. Uhlířová and
Zamastil, 2020 as a singlet instability threshold, where the orbital Hessian
contains a zero eigenvalue with respect to symmetry-pure orbital rotations.
These observations suggest that the RHF approximation somehow breaks down at
$Z\approx 0.84$, but we are not aware of any detailed insight into this
behaviour.

By using the gradient-based GDM algorithm,Van Voorhis and Head-Gordon (2002)
we have accurately converged the RHF ground state for all nuclear charges and
can now firmly establish its properties in the small-$Z$ limit. Remarkably, we
find a sudden increase in $\langle r\rangle$ at
$Z_{\text{c}}^{\text{RHF}}=0.82$ (Fig. 1: middle panel) suggesting that the
RHF approximation can, to a certain extent, represent the ionised system. This
feature closely mirrors the electron dissociation in the UHF wave function,
but gives a less sudden increase. The smoother nature of the RHF dissociation
indicates that the closed-shell restriction on the orbitals artificially
attenuates the electron detachment, providing a less accurate representation
of the exact critical charge than the UHF description.

The RHF electron detachment is accompanied by the onset of another zero
eigenvalue in the orbital Hessian, as described in Ref. Uhlířová and Zamastil,
2020, but we find that this persists for small $Z$ (Fig. 1: bottom panel).
Zero Hessian eigenvalues generally indicate a broken continuous symmetry in
the wave function, such as a global spin-rotation,Burton and Wales (2021); Cui
_et al._ (2013) and define the so-called “Goldstone” manifold of degenerate
states.Cui  _et al._ (2013); Jiménez-Hoyos  _et al._ (2020) In this instance,
the new zero-eigenvalue Hessian mode corresponds to a spin-symmetry-breaking
orbital rotation that also leads to an “inner” and “outer” electron. Since the
energy is constant along this mode, this additional zero Hessian eigenvalue
suggests that the RHF approximation has become unstable with respect to
electron detachment. Consequently, the sudden increase in $\langle r\rangle$
at $Z_{\text{c}}^{\text{RHF}}$ qualitatively represents a closed-shell
critical nuclear charge at $Z_{\text{c}}^{\text{RHF}}=0.82$.

Figure 4: Cumulative radial distribution function for the RHF wave function
using $n=26$. For $Z<0.82$, this function adopts a double-step structure
corresponding to an inner and outer peak in the radial electron density. At
$Z=0.41$, each peak contains half the electron density (dashed line).

To further understand the electron positions in the vicinity of
$Z_{\text{c}}^{\text{RHF}}$, we consider the cumulative radial distribution
function of the doubly-occupied RHF orbital

$N_{\text{RHF}}(r)=\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{r}|\psi_{\text{RHF}}(r^{\prime})|^{2}{r^{\prime}}^{2}\sin\theta\,\mathrm{d}r^{\prime}\,\mathrm{d}\theta\,\mathrm{d}\phi,$
(9)

as shown in Fig. 4. The single-step structure at $Z>Z_{\text{c}}^{\text{RHF}}$
and is consistent with a single peak in the radial distribution function (see
e.g. Fig. 2), indicating that the electrons are closely bound to the nucleus.
For $Z<Z_{\text{c}}^{\text{RHF}}$, this cumulative density adopts a double-
step structure corresponding to a radial density peak close to the nucleus,
and another representing an unbound electron. The magnitude of the second step
continues to grow for smaller $Z$ as the outer peak becomes increasingly
unbound until, at $Z=0.41$, the inner and outer peaks both contain exactly one
electron. For smaller $Z$, all the electron density becomes unbound.
Remarkably, the RHF wave function for $0.41<Z<Z_{\text{c}}^{\text{RHF}}$ is
therefore providing a closed-shell representation of the open-shell atom by
delocalising the electron density over the bound and unbound radial “sites”.
This delocalisation allows the RHF wave function to provide a qualitatively
correct representation of the exact one-body density, but fails to capture any
two-body correlation between the bound and unbound electrons.

###  IV.3 Fractional Spin Error

Although the RHF radial density for $Z<Z_{\text{c}}^{\text{RHF}}$ appears to
be approximating the exact result, the RHF energy remains consistently above
the one-electron hydrogenic energy. The closed-shell nature of the RHF
orbitals means that the inner and outer radial density peaks both contain half
a spin-up electron and half a spin-down electron bound to the nucleus. As a
result, the RHF electron distribution for small $Z$ tends towards a
description of the one-electron atom that also contains half a spin-up and
half a spin-down electron. We have confirmed this limiting behaviour by
computing the RHF energy with a half-occupied orbital, also known as the
“spin-unpolarised” atom with fractional spins.Cohen, Mori-Sánchez, and Yang
(2008a, b); Daas  _et al._ (2020) As expected, this half-occupied RHF solution
becomes degenerate with the two-electron RHF energy at small $Z$ (Fig 1: top
panel).

Remarkably, even though a one-electron atom always has a bound ground state,
we find that the fractional spin RHF wave function predicts an additional
critical nuclear charge at $Z^{\text{frac}}_{\text{c}}=0.41$, where the (half)
electrons suddenly become unbound (Fig 1: middle panel). This critical charge
matches the point where half the electron density has ionised from the nucleus
in the conventional RHF approach (dashed line in Fig. 4). It is well-known
that RHF with fractional spins fails to predict the correct energy for one-
electron atoms, despite the fact that HF theory should be exact in this limit,
and causes the static correlation error that leads to the RHF breakdown for
stretched H2. Cohen, Mori-Sánchez, and Yang (2008a, b) We therefore conclude
that this static correlation also creates an artificial critical nuclear
charge in one-electron atoms at $Z_{\text{c}}^{\text{frac}}=0.41$, and is
responsible for the failure of conventional RHF in the small $Z$ limit.
Identifying similar artificial critical charges using a density functional
approximation would almost certainly provide new insights into the failures of
such methods for anionic energies and electron affinities.Jensen (2010); Kim,
Sim, and Burke (2011); Peach  _et al._ (2015)

##  V Concluding Remarks

In summary, we have used average radial electronic positions to understand
where HF theory predicts electron detachment in the two-electron atom,
providing alternative critical nuclear charges in the RHF and UHF formalisms.
For UHF theory, this critical charge corresponds to a spin-symmetry-breaking
threshold $Z_{\text{c}}^{\text{UHF}}~{}=~{}1.057\,660\,253\,46(1)$ where one
electron suddenly ionises from the nucleus. In contrast, at the RHF critical
charge $Z_{\text{c}}^{\text{RHF}}=0.82$, a secondary peak appears in the
radial distribution function at large distances from the nucleus. These
results provide a broader perspective on electron correlation in the small-$Z$
limit. For example, the RHF $\langle r\rangle$ value starts to deviate from
the exact two-electron result at the UHF symmetry-breaking threshold,
suggesting that $Z_{\text{c}}^{\text{UHF}}$ marks the onset of static
correlation. This static correlation is further supported by the existence of
degenerate UHF solutions representing the dominant configurations in the exact
wave function. Since the UHF radial distribution functions are qualitatively
incorrect for $Z_{\text{c}}<Z<Z_{\text{c}}^{\text{UHF}}$, this static
correlation must be essential for binding the two-electron atom near
$Z_{\text{c}}$. Furthermore, the breakdown of the half-occupied one-electron
RHF result at $Z_{\text{c}}^{\text{frac}}=0.41$ indicates that fractional spin
errors occur for small $Z$, reinforcing the importance of static correlation.

For $Z<Z_{\text{c}}$, the exact wave function contains two dominant resonance
forms with the spin-up or spin-down electron ionised from the nucleus. As the
electrons are indistinguishable, the one-electron density is delocalised
between the bound and unbound “sites”, and this is reflected in the RHF wave
function. However, instantaneous electron-electron correlations ensure that,
when one electron is bound to the nucleus, the other electron becomes unbound.
UHF theory provides a snapshot of these correlations, with one electron
permanently bound to the nucleus, but it cannot describe the resonance between
the two sites. Alternatively, when each orbital can contain a spin-up and
spin-down component in generalised HF (GHF), the symmetry-broken UHF solutions
form a continuum of GHF solutions parameterised by a global spin
rotation.Burton and Wales (2021) The resonance between the two sites is
therefore represented by this continuum, and could be computed using a
(nonorthogonal) linear combination of stationary wave functions.Thom and Head-
Gordon (2009); Burton and Thom (2019)

Finally, the degenerate UHF wave functions form a square-root branch point in
the complex $Z$ plane at $Z_{\text{c}}^{\text{UHF}}$. Following a continuous
complex path around this point interconverts the two degenerate solutions and
swaps the dissociated electron, while a second rotation returns the solutions
to their original forms. We have previously observed this behaviour in
analytic models, but our current results suggest that this phenomenon extends
to the complete-basis-set limit.Burton, Thom, and Loos (2019); Marie, Burton,
and Loos (2020) Furthermore, Ref. Burton, Thom, and Loos, 2019 shows that
these complex branch points can allow a ground-state wave function to be
smoothly “morphed” into an excited-state wave function by following a
continuous complex contour. The two-electron atom therefore provides a new
model for understanding these complex connections near the complete-basis-set
limit, and we intend to continue this investigation in the future.

## Supporting Information

See the supporting information for analytic derivations of the Laguerre-based
one- and two-electron integrals.

## Acknowledgements

I gratefully thank New College, Oxford for funding through the Astor Junior
Research Fellowship. I also thank Hazel Cox for providing the exact two-
electron numerical data from Ref. King  _et al._ , 2015, and Pierre-François
Loos for countless inspiring conversations and critical comments on this
manuscript.

## Data Availability

The data that support the findings of this study are available from the author
upon reasonable request.

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