Asymptotic Analysis of Boundary Layers in a Repulsive Particle System

Abstract

This paper studies the boundary behaviour at mechanical equilibrium at the ends of a finite interval of a class of systems of interacting particles with monotone decreasing repulsive force. This setting covers, for instance, pile-ups of dislocations, dislocation dipoles and dislocation walls. The main challenge in characterising the boundary behaviour is to control the nonlocal nature of the pairwise particle interactions. Using matched asymptotic expansions for the particle positions and rigorous development of an appropriate energy via $\Gamma$-convergence, we obtain the equilibrium equation solved by the boundary layer correction, associate an energy with an appropriate scaling to this correction, and provide decay rates into the bulk.

Appendices

Appendix A: Technical Details of Asymptotic Matching

1.1 A.1 Asymptotic Analysis Using the Bulk Ansatz

We obtain asymptotic solutions for $x(i)$ in both the bulk and boundary layer regimes using the method of matched asymptotic expansions. The method that we use involves matching with an intermediate variable, and is analogous to the methods used in [20, 43]. Whereas [20, 43] concentrate on leading-order matching, we use the method to obtain higher order corrections. We begin by introducing a continuum bulk ansatz, $x(i;n) = \xi (in^{-1};n)$, which we assume to be valid when $i \gg 1$ and $n-i \gg 1$. At the same time, we introduce a discrete boundary layer ansatz, $x(i;n) = n^{-1} \chi (i;n)$, which we assume to be valid when $i \ll n$. Thus, both ansatzes are assumed to be valid asymptotic expansions when $1 \ll i \ll n$.

This means that we can introduce an arbitrary $K$ with $1 \ll K \ll n$ and use the boundary layer ansatz for $x(i;n)$ when $i \leq K$ or $i \geq n- K$ and use the bulk ansatz for $x(i;n)$ when $K < i < n-K$. By the principles that underly the method of matched asymptotic expansions, the precise dependence of $K$ on $n$ should not matter; the behaviour of $\xi (s;n)$ as $s \to 0$ should match with the behaviour of $\chi (i;n)$ as $i \to \infty$ so as to yield consistent asymptotic expressions for $x(i)$ when $1 \ll i \ll n$ regardless of whether $i$ is treated as being in the bulk regime or the boundary layer regime. We think of $K$ as an arbitrary intermediate point where we connect the bulk ansatz with the boundary layer ansatz.

We make the following assumptions about the behaviour of $\xi (s;n)$ and $\chi (i;n)$:

(MinSpacing) :

There exists $M > 0$ such that $\chi (i+1) - \chi (i) \geq M$ whenever $i \ll n$, and $\xi '(s) \geq M$ whenever $s \gg n^{-1}$ and $1 - s \gg n^{-1}$.

( $\xi$ -Smooth) :

$\xi \in C^{\infty }((\eta ,1-\eta ))$ for any choice of $\eta$ where $n^{-1} \ll \eta \ll 1$.

As previously, these statements must all hold true in the asymptotic limit as $n \to \infty$ where $\xi$ and $\chi$ are replaced with

$$ \xi (s) = \sum_{k=0}^{Q} n^{-b_{k}} \xi _{k}(s) + O \bigl(n^{-b_{Q+1}} \bigr), \quad \text{and} \quad \chi (i) = \sum_{j=0}^{P} n^{-\beta _{j}} \chi _{j}(i) + O \bigl(n^{-\beta _{P+1}} \bigr), $$

respectively for any choices of $P$ and $Q$.

We note that (MinSpacing) implies that $\chi (i+1) - \chi (i) \geq M$ whenever $i \leq K$, and equally that

$$ n \int _{\frac{i}{n}}^{\frac{i+1}{n}} \xi (s) \mathrm{d} s \geq M, $$

whenever $i \geq K$, regardless of the choice of $K$ as long as $1 \ll K \ll n$. Hence, (MinSpacing) implies a minimum separation between particles that holds uniformly in $n$ independently of the choice of ‘cutoff’ between the bulk region and the boundary layer region.

We also note that replacing $\chi$ and $\xi$ with their leading order approximations in (MinSpacing) and considering the limits as $n \to \infty$, yields the result that $\chi _{0}(i+1) - \chi _{0}(i) \geq M$ and $\xi _{0}'(s) \geq M$ throughout. Additionally, we observe that (MinSpacing) places growth restrictions on higher order corrections to $\chi$ and $\xi '$. Specifically, it means that $\chi _{j}(i+1) - \chi _{j}(i)$ cannot grow (negatively) at a rate greater than $i^{\beta _{j}}$ as $i \to \infty$, and that $\xi _{k}'(s)$ cannot grow (negatively) at a rate greater than $s^{-b_{k}}$ as $s \to 0$.

We further use the symmetry of the problem to assert that $\xi (1-s) = 1-\xi (s)$ and that $x(n-i) = 1 - n^{-1} \chi (n-i;n)$ when $n - i = O(1)$. We also assume that the bulk ansatz and the discrete boundary layer ansatz are the only scalings that we need to consider for the method of matched asymptotic expansions. That is, we assume that there is no distinguished intermediate scaling between $i = \operatorname{ord}(n)$ and $i = \operatorname{ord}(1)$. A justification of this assumption can be obtained by using the methods described in Sect. 2.3.

Given that $V(x) = |x|^{-a}$, the force balance equation from (2) yields

$$ an^{-a-1} \Biggl( \sum_{k=1}^{i} \bigl[x(i) - x(i-k) \bigr]^{-a-1} - \sum_{k=1}^{n-i} \bigl[x(i+k) -x(i) \bigr]^{-a-1} \Biggr) =0. $$

(54)

In the remainder of this section, we concentrate on analysing force balance in the bulk, where $i = \operatorname{ord}(n)$. As described above, we split the sums into regions where we apply the continuum ansatz for $x(i\pm k)$ and regions where we apply the discrete ansatz:

$$\begin{aligned} &\underbrace{an^{-a-1} \sum_{k=0}^{K-1} \bigl( \bigl[\xi \bigl(in^{-1} \bigr) - n^{-1}\chi (k) \bigr]^{-a-1} - \bigl[ 1 - \xi \bigl(in^{-1} \bigr) - n^{-1}\chi (k) \bigr]^{-a-1} \bigr) }_{S_{0}} \\ &\quad {}+ an^{-a-1} \Biggl( \sum_{k=1}^{i-K} \bigl[\xi \bigl(in^{-1} \bigr) - \xi \bigl([i-k]n^{-1} \bigr) \bigr]^{-a-1} \\ &\quad {}- \sum_{k=1}^{n-K-i} \bigl[\xi \bigl([i+k]n^{-1} \bigr) -\xi \bigl(in^{-1} \bigr) \bigr]^{-a-1} \Biggr) =0. \end{aligned}$$

(55)

Since $K \ll n$, the sum marked $S_{0}$ in (55) is $o(n^{-a})$. From previously, we recognise that the leading order terms in the bulk force balance will be $\operatorname{ord}(n^{-1})$; hence, it will be possible to obtain expressions for $\xi (s)$ up to $o(n^{-(a-1)})$ while entirely neglecting any contributions from $S_{0}$. Higher order corrections to $\xi (s)$ may be obtained by expanding the summand of $S_{0}$ using Taylor series, and then exploiting the properties of $\chi (i)$. While it is possible to carry out these manipulations, we do not consider these high-order corrections in detail in this paper.

We can therefore follow the approach used previously and neglect $S_{0}$. This leads us to define the following force function, $F(s)$, noting that force balance in the bulk requires $F(s) = o(n^{-a})$ for all $s = \frac{i}{n}$ where $i = \operatorname{ord}(n)$ and $n-i = \operatorname{ord}(n)$:

$$ F(s) := an^{-a-1} \Biggl[ \sum_{k=1}^{ \lfloor sn -K \rfloor } \bigl[\xi (s) - \xi \bigl(s-kn^{-1} \bigr) \bigr]^{-a-1} - \sum _{k=1}^{ \lfloor n-K-sn \rfloor } \bigl[\xi \bigl(s+kn^{-1} \bigr) -\xi (s) \bigr]^{-a-1} \Biggr]. $$

We now separate $F(s)$ into three parts as previously, introducing an arbitrary integer $H$ where $n^{\frac{a}{a+1}} \ll H \ll n$:

$$\begin{aligned} F(s) =& \underbrace{a n^{-a-1}\sum _{k=1}^{H} \bigl( \bigl[\xi (s) - \xi \bigl(s-kn^{-1} \bigr) \bigr]^{-a-1} - \bigl[\xi \bigl(s+kn^{-1} \bigr) -\xi (s) \bigr]^{-a-1} \bigr)}_{=: S_{1}} \\ & {}+ \underbrace{a n^{-a-1} \sum_{k=H+1}^{ \lfloor sn \rfloor - K} \bigl[\xi (s) - \xi \bigl(s-kn^{-1} \bigr) \bigr]^{-a-1}}_{=: S_{2}} \\ & {}- \underbrace{a n^{-a-1} \sum_{k=H+1}^{ \lfloor n-sn \rfloor - K} \bigl[\xi \bigl(s+kn^{-1} \bigr) -\xi (s) \bigr]^{-a-1}}_{=: S_{3}}. \end{aligned}$$

(56)

We note that $H \gg n^{\frac{a}{a+1}}$ places restrictions on $K$, since we require $K \gg H$ in order for the sums $S_{2}$ and $S_{3}$ to contain large numbers of terms. This lower bound on $K$ might suggest the presence of a distinguished scaling between $i = \operatorname{ord}(n)$ and $i = \operatorname{ord}(1)$, so that there is a continuum boundary layer problem to solve between the continuum bulk problem and the discrete boundary layer. We expect that the methods in Sect. 2.3 could be used to show that no such continuum boundary layer problem can exist and that hence the bulk ansatz is valid for all $i \gg 1$, but we do not pursue this analysis further.

We begin our analysis of (56) by considering $S_{2}$. Using the Euler–Maclaurin summation formula with an offset from the integers (see, for example, [39]), we find that

$$\begin{aligned} S_{2} =& an^{-a} \int _{Kn^{-1}}^{s-Hn^{-1}} \frac{\mathrm{d} u}{[\xi (s) - \xi (u) ]^{a+1}} \\ & {}- an^{-a-1} \biggl( \frac{1}{2 [\xi (s) - \xi (s-Hn^{-1})]^{a+1}} + \frac{B_{1}(\{sn\})}{[\xi (s) - \xi (Kn^{-1})]^{a+1}} \biggr) \\ & {}+ a(a+1) n^{-a-1} \int _{Kn^{-1}}^{s-Hn^{-1}} \frac{B_{1}(\{sn+un\}) \xi '(u)}{[\xi (s) - \xi (u) ]^{a+2}} \mathrm{d}u, \end{aligned}$$

(57)

where $B_{1}(\{\cdot \})$ is the 1-periodic extension of the first Bernoulli polynomial. Using (MinSpacing) we observe that $\xi (s) - \xi (s-Hn^{-1}) \geq MHn^{-1}$ and that $\xi (s) - \xi (Kn^{-1}) = \operatorname{ord}(1)$. Using Hölder’s inequality to show that the integral remainder term is asymptotically no larger than the terms on the second line of (57), we therefore find that

$$ S_{2} = an^{-a} \int _{Kn^{-1}}^{s-Hn^{-1}} \frac{\mathrm{d} u}{[\xi (s) - \xi (u) ]^{a+1}} + O \bigl(H^{-a-1} \bigr). $$

An identical argument applies to $S_{3}$. Using the fact that $H \gg n^{\frac{a}{1+a}}$ and $K \ll n$, we can combine the expansions of $S_{2}$ and $S_{3}$ to show that

$$ S_{2} + S_{3} = an^{-a} \biggl[ \int _{0}^{s-Hn^{-1}} \frac{\mathrm{d}u}{[\xi (s) - \xi (u)]^{a+1}} - \int _{s+Hn^{-1}}^{1} \frac{\mathrm{d}u}{[\xi (u) - \xi (s)]^{a+1}} \biggr] + o \bigl(n^{-a} \bigr). $$

(58)

Now consider $S_{1}$. Using Taylor’s theorem, ( $\xi$ -Smooth) implies that $\xi (s\pm kn^{-1})$ can be approximated by the series

$$ \xi \bigl(s\pm kn^{-1} \bigr) \sim \xi (s) \pm \xi '(s) kn^{-1} + \frac{1}{2} \xi ''(s) k^{2}n^{-2} + \cdots , $$

which is asymptotic for any $k \ll n$ and $s \gg n^{-1}$. Since $k \leq H \ll n$ in $S_{1}$ and $s = \operatorname{ord}(1)$ in our present analysis, we can apply this asymptotic expansion throughout. This yields

$$\begin{aligned} S_{1} \sim & a \sum_{k=1}^{H} k^{-a-1} \biggl( \biggl[\xi '(s) - \frac{1}{2} k n^{-1}\xi ''(s) + \frac{1}{6} k^{2} n^{-2} \xi '''(s) + \cdots \biggr]^{-a-1} \\ & {}- \biggl[\xi '(s) + \frac{1}{2} k n^{-1} \xi ''(s) + \frac{1}{6} k^{2} n^{-2} \xi '''(s) + \cdots \biggr]^{-a-1} \biggr). \end{aligned}$$

(59)

Noting that $k n^{-1} \ll 1$, we can apply the binomial series and rearrange to obtain

$$\begin{aligned} S_{1} \sim& \sum_{k=1}^{H} k^{-a-1} \biggl( (-a)_{2} \xi ''(s) \bigl[\xi '(s) \bigr]^{-a-2} k n^{-1} + \biggl[ \frac{1}{12} (-a)_{2} \bigl[\xi '(s) \bigr]^{-a-2} \xi ''''(s) \\ & {}+ \frac{1}{6}(-a)_{3} \bigl[\xi '(s) \bigr]^{-a-3} \xi ''(s) \xi '''(s) + \frac{1}{24} (-a)_{4} \bigl[\xi '(s) \bigr]^{-a-4} \bigl[\xi ''(s) \bigr]^{3} \biggr] k^{3} n^{-3} \\ & {}+ O \bigl(k^{5} n^{-5} \bigr) \biggr) \end{aligned}$$

(60)

where $(\cdot )_{r}$ is the Pochhammer symbol, defined so that $(\alpha )_{r} := \alpha \cdot (\alpha - 1) \cdots (\alpha - r + 1)$. Since the summand in Eq. (60) is obtained by taking compositions of functions defined as formal series, we can use the properties of partial Bell polynomials (see, for example, [11, 45]) to obtain a general expression for the terms in the summand of (60). Specifically, we find that

$$ S_{1} \sim \sum_{k=1}^{H} \sum_{p=0}^{\infty }\mathscr{B}_{p}[ \xi ](s) k^{-a+2p} n^{-(2p+1)}, $$

(61)

where

$$\begin{aligned} &\mathscr{B}_{p}[\xi ](s) \\ &\quad := \Biggl( \frac{2}{(2p+1)!} \sum_{q=1}^{2p+1}Y_{2p+1,q} \biggl[ \frac{\xi ''(s)}{2},\frac{\xi '''(s)}{3},\ldots ,\frac{\xi ^{(2p-q+3)}}{2p-q+3} \biggr] (-a)_{q+1} \bigl[\xi '(s) \bigr]^{-a-1-q} \Biggr) , \end{aligned}$$

(62)

and $Y_{p,q}(t_{1},t_{2},\ldots ,t_{p-q+1})$ is a partial Bell polynomial. These polynomials are defined by the expression

$$ Y_{p,q}(t_{1},t_{2},\ldots ,t_{p-q+1}) = \sum \frac{p!}{r_{1}! r_{2}! \ldots r_{p-q+1}!} \biggl( \frac{t_{1}}{1!} \biggr) ^{r_{1}} \biggl( \frac{t_{2}}{2!} \biggr) ^{r_{2}} \ldots \biggl( \frac{t_{p-q+1}}{(p-q+1)!} \biggr) ^{r_{p-q+1}}, $$

where the sum is taken over all integer sequences $\{r_{1}, r_{2}, \ldots , r_{p-q+1}\}$ where

$$ \sum_{k=1}^{p-q+1} k r_{k} = p, \quad \text{and} \quad \sum_{k=1}^{p-q+1} r_{k} = q. $$

As described in [11], partial Bell polynomials have the property that

$$ \frac{1}{q!} \Biggl[ \sum_{j=1}^{\infty } \frac{x^{j}}{j!} t_{j} \Biggr] ^{q} = \sum _{p=q}^{\infty } Y_{p,q}(t_{1},t_{2}, \ldots ,t_{p-q+1}) \frac{x^{p}}{p!}. $$

It is this property of partial Bell polynomials that makes it possible to obtain (62) from (59).

Since $kn^{-1} \ll 1$, it is possible to swap the order of summation in (61) while still retaining asymptoticity:

$$ S_{1} \sim \sum_{p=0}^{\infty } \mathscr{B}_{p}[\xi ](s) n^{-(2p+1)} \sum _{k=1}^{H} k^{-a+2p}. $$

To evaluate the sum over $k$, we note from [20] that the asymptotic behaviour of the generalised harmonic numbers is given by

$$ \sum_{k=1}^{H} k^{-r} = \textstyle\begin{cases} \zeta (r) - \frac{1}{r-1} {H^{-(r-1)}} + O(H^{-r}), & r > 1, \\ \log (H) + \gamma + O(H^{-1}), & r = 1, \\ O(H^{1-r}), & r < 1, \end{cases} $$

where $\gamma$ is the Euler–Mascheroni constant. Then, from $n^{\frac{a}{a+1}} \ll H \ll n$ we obtain

$$ \sum_{k=1}^{H} k^{-a+2p} n^{-(2p+1)} = \textstyle\begin{cases} \zeta (a-2p)n^{-(2p+1)} - \frac{(Hn^{-1})^{-(a-2p-1)}}{a-2p-1}n^{-a} + o(n^{-a}), & 2p < a-1, \\ ( \log (H) + \gamma ) n^{-a} + o(n^{-a}) & 2p = a-1, \\ o(n^{-a}), & 2p > a-1, \end{cases} $$

and (61) yields

$$\begin{array}{rcl}{S}_1& =& \sum _{p=0}^{\lceil \frac{a-1}{2}\rceil -1}({\mathcal{B}}_p[\xi ](s)[\zeta (a-2p)n^{-(2p+1)}-\frac{{(H{n}^{-1})}^{-(a-2p-1)}}{a-2p-1}{n}^{-a}])\\ & & +{\mathbb{1}}_{a\in 2\mathbb{N}+1}{\mathcal{B}}_{\frac{a-1}{2}}[\xi ](s)[log(H)+\gamma ]n^{-a}+o\left(n^{-a}\right).\end{array}$$

(63)

In order to simplify this expression into a form where it can be combined with (58), it is useful to introduce finite part integration. Following [28, 32], we define the one-sided finite part integral for functions that are well-behaved apart from a possible singularity at zero, and which satisfy

$$ \psi (u) = \sum_{j=0}^{\Upsilon -1} c_{j} u^{-a_{j}} + c_{\Upsilon } u^{-1} + O \bigl(u^{-1+\delta } \bigr), \quad \text{as }u \to 0, $$

where $a_{0} > a_{1} > \cdots > a_{\Upsilon -1} > 1$ and $\delta > 0$. In this case, we define

$$\underset{0}{\overset{y}{⨍}}\psi (u)\mathrm{d}u:=\underset{\eta \to 0}{lim}[{\int }_{\eta }^y\psi (u)\mathrm{d}u-\sum _{j=0}^{\mathrm{\Upsilon }-1}\frac{c_j{\eta }^{-(a_j-1)}}{a_j-1}-c_{\mathrm{\Upsilon }}log\frac{1}{\eta }].$$

(64)

From (61), we note that

$$ a \bigl[\xi (s) - \xi (s-u) \bigr]^{-a-1} - a \bigl[\xi (s+u) -\xi (s) \bigr]^{-a-1} = \sum_{p=0}^{ \lfloor \frac{a-1}{2} \rfloor } \mathscr{B}_{p}[\xi ](s) u^{-(a-2p)} + O \bigl( u^{1 + 2 \lfloor \frac{a}{2} \rfloor - a} \bigr) . $$

Then, using (64), we obtain

$$\begin{array}{rcl}& & \underset{0}{\overset{y}{⨍}}a{[\xi (s)-\xi (s-u)]}^{-a-1}-a{[\xi (s+u)-\xi (s)]}^{-a-1}\mathrm{d}u\\ & & =-\sum _{p=0}^{\lceil \frac{a-1}{2}\rceil -1}{\mathcal{B}}_p[\xi ](s)\frac{y^{-(a-2p-1)}}{a-2p-1}+{\mathbb{1}}_{a\in 2\mathbb{N}+1}{\mathcal{B}}_{\frac{a-1}{2}}[\xi ](s)log(y)+o(1)\end{array}$$

as $y \rightarrow 0$. Hence, (63) becomes

$$\begin{array}{rcl}{S}_1& =& \sum _{p=0}^{\lceil \frac{a-1}{2}\rceil -1}{\mathcal{B}}_p[\xi ](s)\zeta (a-2p)n^{-(2p+1)}\\ & & +a{n}^{-a}\underset{0}{\overset{H{n}^{-1}}{⨍}}{[\xi (s)-\xi (s-u)]}^{-a-1}-{[\xi (s+u)-\xi (s)]}^{-a-1}\mathrm{d}u\\ & & +{\mathbb{1}}_{a\in 2\mathbb{N}+1}{\mathcal{B}}_{\frac{a-1}{2}}[\xi ](s)[log(n)+\gamma ]n^{-a}+o\left(n^{-a}\right).\end{array}$$

Combining with (58), we therefore find that

$$\begin{array}{rcl}F(s)& =& \sum _{p=0}^{\lceil \frac{a-1}{2}\rceil -1}{\mathcal{B}}_p[\xi ](s)\zeta (a-2p)n^{-(2p+1)}\\ & & +({\mathbb{1}}_{a\in 2\mathbb{N}+1}{\mathcal{B}}_{\frac{a-1}{2}}[\xi ](s)[log(n)+\gamma ]+a\underset{0}{\overset{1}{⨍}}\frac{sgn(s-u)}{{|\xi (s)-\xi (u)|}^{a+1}}\mathrm{d}u)n^{-a}+o\left(n^{-a}\right).\end{array}$$

(65)

In the more general case where $V$ satisfies (30), we find that much of the argument outlined in this section still holds. Since it is possible to approximate $V'(x)$ by $-ax^{-a-1}$ for large $x$, we find that $S_{2}+S_{3}$ will still be given by (58). The most significant changes required to generalise our argument involve the manipulation of $S_{1}$. Repeated use of Taylor series (analogous to the manipulations of $V$ in Sect. 2.2) are needed to obtain a new definition for $\mathscr{B}_{p}$ for a general $V$; specifically, we find that $(-a)_{q+1} [\xi '(s)]^{-a-1-q}$ in (62) should be replaced with $k^{a+1+q}V^{(q+1)}[\xi '(s) k]$.

While it is true that

$$ k^{a+1+q}V^{(q+1)} \bigl[\xi '(s) k \bigr] \to (-a)_{q+1} \bigl[\xi '(s) \bigr]^{-a-1-q} \quad \text{as } k \to \infty , $$

the fact that the modified definition of $\mathscr{B}_{p}$ involves $k$ creates complications for the manipulation of sums involving $k$ through the rest of the argument. Ultimately, we find that the asymptotic properties of these sums mean that the approach outlined above remains valid, and that the analogous equation to (65) is

$$\begin{array}{rcl}F(s)& =& \sum _{p=0}^{\lceil \frac{a-1}{2}\rceil -1}{\overline{\mathcal{B}}}_p[\xi ](s)n^{-(2p+1)}\\ & & +({\mathbb{1}}_{a\in 2\mathbb{N}+1}[{\tilde{\mathcal{B}}}_{\frac{a-1}{2}}[\xi ](s)log(n)+\mathcal{G}[\xi ]]+a\underset{0}{\overset{1}{⨍}}\frac{sgn(s-u)}{{|\xi (s)-\xi (u)|}^{a+1}}\mathrm{d}u)n^{-a}\\ & & +o\left(n^{-a}\right),\end{array}$$

where $\bar{\mathscr{B}}_{p}$, $\tilde{\mathscr{B}}_{\frac{a-1}{2}}$, and $\mathscr{G}$ depend on $V$. Note that the terms which gave rise to the zeta function and Euler–Mascheroni constant in (65) are replaced with new formulations that depend on $V$ and $\xi '$, but the overall structure of the total force from (65) remains the same. We find that

$$ \bar{\mathscr{B}}_{0}[\xi ](s) = \xi ''(s) \sum_{k=1}^{\infty } V'' \bigl[\xi '(s) k \bigr] k^{2}, $$

(66)

and that $\bar{\mathscr{B}}_{p}[\xi ]$, $\tilde{\mathscr{B}}_{\frac{a-1}{2}}[\xi ]$ and $\mathscr{G}[\xi ]$ all evaluate to the zero function when $\xi$ is affine. These observations enable us to extend the results of the following section to more general potentials $V$ that satisfy (30).

1.2 A.2 Solving for Higher Order Corrections in the Bulk

We now return to the case where $V(x) = |x|^{-a}$ and we seek an asymptotic expansion of $\xi (s)$ that will enable (55) to be satisfied for integers $i$ where $i \gg K$ and $n-i \gg K$. If we restrict our analysis to corrections up to $\operatorname{ord} [n^{-(a-1)} ]$, we find that this is equivalent to seeking $\xi (s)$ so that $F(s) = o(n^{-a})$, and hence we can make immediate use of (65). Thus, we begin by expanding $\xi (s)$ as an asymptotic series as follows:

$$ \xi (s) = \xi _{0}(s) + \sum _{k=1}^{\bar{Q}} n^{-b_{k}} \xi _{k}(s) + n^{-(a-1)} \tilde{\xi }(s) + o \bigl(n^{-(a-1)} \bigr), \quad 0 < b_{1} < \cdots < b_{\bar{Q}} < a-1; $$

(67)

where $\bar{Q}$ may perhaps be infinite or zero.

On substituting (67) into (65), we find that the largest nontrivial terms are recovered at $O(n^{-1})$. These yield the result that $\zeta (a)\mathscr{B}_{0}[\xi _{0}](s) = 0$. Using the definition of $\mathscr{B}_{p}[\xi ]$ in (62), this becomes

$$ \zeta (a) (-a)_{2} \xi _{0}''(s) \bigl[\xi _{0}'(s) \bigr]^{-a-2} = 0, $$

and hence $\xi _{0}(s)$ is affine. More specifically, we can use the leading order boundary conditions from (15) to conclude that $\xi _{0}'(s) = 1$ and $\xi _{0}(s) = s$.

In order to characterise the next nontrivial term in the expansion of $F(s)$, we assume for the moment that $b_{1} < a-1$ to avoid dealing with the singular integral term at $O(n^{-a})$. Since $\xi _{0}'(s)$ is constant and nonzero, it follows from (62) that $\mathscr{B}_{p}[\xi _{0}] \equiv 0$ for all $p$. Hence, the next nontrivial terms in the expansion of (65) appear at $O(n^{-1-b_{1}})$, where we find that

$$ \zeta (a) (-a)_{2} \xi _{1}''(s) = 0. $$

Again, we conclude that $\xi _{1}(s)$ is affine and we find that $\mathscr{B}_{p}[\xi _{0} + n^{-b_{1}}\xi _{1}] \equiv 0$ for all $p$. We cannot apply boundary conditions to $\xi _{1}(s)$ at this stage, since the boundary conditions on $\xi _{1}$ will depend on the matching between the bulk solution and the boundary layer solution. However, we can use the symmetry of the force balance problem to conclude that $\xi _{1}(s) = -\xi _{1}(1-s)$ and hence

$$ \xi _{1}(s) = \biggl(s - \frac{1}{2} \biggr) p_{1} $$

where $p_{1} = \xi _{1}'(s)$ is a constant to be determined from matching with the boundary layer.

As long as $b_{k} < a-1$ we can apply the same argument to show that $\xi _{k}$ is affine. We will use this freedom in the choice of $b_{k}$ later on to match with the boundary layer. For now, we rewrite the expansion of $\xi$ in (67) as

$$ \xi (s) = s + \bar{p}(n) \biggl(s - \frac{1}{2} \biggr) + n^{-(a-1)} \tilde{\xi }(s) + o \bigl(n^{-(a-1)} \bigr) $$

where $\bar{p} := p_{1} n^{-b_{1}} + p_{2} n^{-b_{2}} + \cdots$, so that $\bar{p} \ll 1$. This yields

$$n^{-a}[\zeta (a){(-a)}_2{\tilde{\xi }}^{″}(s)+a\underset{0}{\overset{1}{⨍}}\frac{sgn(s-u)}{{|s-u|}^{a+1}}\mathrm{d}u]=o\left(n^{-a}\right).$$

Next we solve for $\tilde{\xi }$. Since

$$a\underset{0}{\overset{1}{⨍}}\frac{sgn(s-u)}{{|s-u|}^{a+1}}\mathrm{d}u={(1-s)}^{-a}-s^{-a},$$

we have that

$$ \tilde{\xi }''(s) = \frac{s^{-a} - (1-s)^{-a}}{\zeta (a) (-a)_{2}}. $$

By using again the symmetry of the force balance (i.e. $\tilde{\xi }(s) = - \tilde{\xi }(1-s)$), we obtain

$$ \tilde{\xi }(s) = \left\{ \textstyle\begin{array}{l@{\quad }l} \frac{s^{-(a-2)} - (1-s)^{-(a-2)}}{\zeta (a) (-a+2)_{4}} + \biggl(s - \frac{1}{2} \biggr)\tilde{p} , & a \neq 2 \\ \frac{\log (1-s) - \log (s) }{\pi ^{2}} + \biggl(s - \frac{1}{2} \biggr)\tilde{p} , & a = 2. \end{array}\displaystyle \right. $$

(68)

where $\tilde{p}$ is a constant to be determined from matching with the boundary layer.

In the more general case where $V$ satisfies (30), we still find that $\xi _{k}(s) = p_{k}(s - \frac{1}{2})$ whenever $b_{k} < a-1$ as a consequence of the fact that $\bar{\mathscr{B}}_{p}[\xi ] \equiv 0$ when $\xi$ is affine. We can also evaluate $\tilde{\xi }$ by using the definition of $\bar{\mathscr{B}}_{0}$ given in (66). This yields

$$ \tilde{\xi }(s) = \left\{ \textstyle\begin{array}{l@{\quad }l} \frac{s^{-(a-2)} - (1-s)^{-(a-2)}}{Z(V) (-a+2)_{2}} + \biggl(s - \frac{1}{2} \biggr)\tilde{p} , & a \neq 2 \\ \frac{\log (1-s) - \log (s) }{Z(V)} + \biggl(s - \frac{1}{2} \biggr)\tilde{p} , & a = 2. \end{array}\displaystyle \right. $$

(69)

where

$$ Z(V) := \sum_{k=1}^{\infty } V''(k) k^{2}. $$

1.3 A.3 Asymptotic Analysis in the Boundary Layer

We now return to assuming $V(x) = |x|^{-a}$ and seek solutions for $\chi _{j}(i)$ by considering the case where $i = \operatorname{ord}(1)$ in (54). We recall that we introduced $K$ at the beginning of Sect. A.1 so that $1 \ll K \ll n$, and hence $K$ is in the intermediate region where both the boundary layer ansatz and the bulk ansatz can be used.

Assuming $i = \operatorname{ord}(1)$, we split the sums in (54) to obtain

$$\begin{aligned} &\underbrace{a \sum_{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{K} \frac{\operatorname{sgn}(i-k)}{|\chi (i) - \chi (k)|^{a+1}}}_{S_{4}} -\underbrace{an^{-a-1}\sum _{k=K+1}^{n-K-1} \bigl[\xi \bigl(kn^{-1} \bigr) -n^{-1}\chi (i) \bigr]^{-a-1}}_{S_{5}} \\ &\quad {}-\underbrace{an^{-a-1}\sum_{k=0}^{K} \bigl[1-n^{-1}\chi (k) -n^{-1}\chi (i) \bigr]^{-a-1}}_{S_{6}} =0. \end{aligned}$$

(70)

Since all the terms in the summand of $S_{6}$ are $O(1)$, we find that $S_{6} = O(Kn^{-a-1}) = o(n^{-a})$. Moreover, using (MinSpacing) we obtain that

$$ S_{5} \lesssim n^{-a-1} \sum_{k=K+1}^{n-K-1} \bigl[M kn^{-1} \bigr]^{-a-1} \lesssim \sum _{k=K}^{\infty } k^{-a-1} = O \bigl(K^{-a} \bigr), $$

and hence (70) becomes

$$ a \sum_{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{K} \operatorname{sgn}(i-k)|\chi (i) - \chi (k)|^{-a-1} = O \bigl(K^{-a} \bigr). $$

(71)

Following the methods described in Sect. 2.4, it follows that the leading order solution in the boundary layer is a solution to the infinite system of algebraic equations

$$ a \sum_{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{\infty } \operatorname{ \text{sgn}}(i-k)|\chi _{0}(i) - \chi _{0}(k)|^{-a-1} = 0, \quad i = 1, 2, \ldots $$

subject to the matching condition $\chi _{0}(i) - \chi _{0}(i-1) \rightarrow 1$ as $i \rightarrow \infty$.

To obtain higher order corrections, we begin by assuming an asymptotic power series expansion for $\chi (i)$. As in Sect. A.2, we will seek solutions up to $\operatorname{ord}[n^{-(a-1)}]$ and thus it is convenient to introduce a power series of the form

$$ \chi (i) = \chi _{0}(i) + \sum _{j=1}^{\bar{P}} n^{-\beta _{j}} \chi _{j}(i) + n^{-(a-1)} \tilde{\chi }(i) + o \bigl[n^{-(a-1)} \bigr], \quad 0 < \beta _{1} < \cdots < \beta _{\bar{P}} < a-1; $$

(72)

where $\bar{P}$ may be zero or infinite.

As we discuss in Sect. A.4, asymptotic matching implies that $\chi _{j}(i) - \chi _{j}(i-1)$ must have a finite limit as $i \to \infty$ for any $\beta _{j} < a-1$, and an identical result holds for $\tilde{\chi }(i) - \tilde{\chi }(i-1)$. The fact that these limits are finite enables us to make significant simplifications after we substitute (72) into (71). Applying the multinomial expansion, we see that this yields

$$\begin{aligned} &a \sum_{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{K} \operatorname{sgn}(i-k)|\chi (i) - \chi (k)|^{-a-1} \sim a \sum _{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{K} \operatorname{sgn}(i-k)| \chi _{0}(i) - \chi _{0}(k)|^{-a-1} \\ &\qquad {}- n^{-\beta _{1}} a(a+1) \sum_{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{K} |\chi _{0}(i) - \chi _{0}(k)|^{-a-2} \bigl[ \chi _{1}(i) - \chi _{1}(k) \bigr] \\ &\qquad {}- n^{-\beta _{2}} a(a+1) \sum_{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{K} |\chi _{0}(i) - \chi _{0}(k)|^{-a-2} \bigl[ \chi _{2}(i) - \chi _{2}(k) \bigr] \\ &\qquad {}+ \frac{n^{-2\beta _{1}}}{2} a(a+1) (a+2) \sum_{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{K} |\chi _{0}(i) - \chi _{0}(k)|^{-a-3} \bigl[ \chi _{1}(i) - \chi _{1}(k) \bigr] ^{2} + \cdots \\ &\quad = O \bigl(K^{-a} \bigr). \end{aligned}$$

(73)

Since $\chi _{j}(i) - \chi _{j}(k) \sim C_{j} (i- k)$ for some constant $C_{j}$ as $k \to \infty$, it follows that

$$\begin{aligned} &\sum_{k=K+1}^{\infty } \operatorname{ \text{sgn}}(i-k)|\chi _{0}(i) - \chi _{0}(k)|^{-a-1} = O \bigl(K^{-a} \bigr), \\ &\sum_{k=K+1}^{\infty } |\chi _{0}(i) - \chi _{0}(k)|^{-a-2} \bigl[ \chi _{1}(i) - \chi _{1}(k) \bigr] = O \bigl(K^{-a} \bigr), \end{aligned}$$

and so on. This enables us to extend the sums in (73) to infinity without introducing significant errors. Choosing $K$ so that $n^{1-\frac{1}{a}} \ll K \ll n$, we therefore find that (71) becomes

$$\begin{aligned} &a \sum_{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{\infty } \operatorname{ \text{sgn}}(i-k)|\chi _{0}(i) - \chi _{0}(k)|^{-a-1} \\ &\quad {}- n^{-\beta _{1}} a(a+1) \sum_{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{\infty } |\chi _{0}(i) - \chi _{0}(k)|^{-a-2} \bigl[ \chi _{1}(i) - \chi _{1}(k) \bigr] + \cdots = o \bigl[n^{-(a-1)} \bigr]. \end{aligned}$$

Collecting $\operatorname{ord}(n^{-\beta _{1}})$ terms, we obtain the following infinite homogeneous linear system for system for $\chi _{1}(i)$:

$$ -a(a+1) \sum_{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{\infty } | \chi _{0}(i) - \chi _{0}(k)|^{-a-2} \bigl[ \chi _{1}(i) - \chi _{1}(k) \bigr] = 0, \quad i = 1, 2, \ldots , $$

(74)

where $\chi _{1}(0) = 0$ due to the fact that $x(0) = 0$. This system must be solved subject to some matching condition that relates the behaviour of $\chi _{1}(i)$ as $i \to \infty$ to the behaviour of the bulk solution as $s \to 0$. Since $|\chi _{0}(i) - \chi _{0}(k)|^{-a-2} = O(k^{-a-2})$ as $k \to \infty$, we observe that the sums in (74) are absolutely convergent when $\chi _{1}(k) = O(k^{a+1-\delta })$ for some $\delta > 0$. Since asymptotic matching gives $\chi _{j}(k) = O(k)$ as $k \to \infty$ for all $\beta _{j} < a-1$, it follows that the sums in (74) are absolutely convergent for any $i$.

In Sect. A.4, we show that asymptotic matching can be used to determine the exponents $\beta _{j}$ and $b_{k}$. As we will see, this analysis relies on the claim that if (74) is solved subject to the particular matching condition $\chi _{1}(i) - \chi _{1}(i-1) \to 0$, then the only possible solution is the trivial solution, $\chi _{1}(i) \equiv 0$. To prove this claim, we observe that (74) is a linear equation of the type $\mathscr{A} \chi _{1} = 0$, where we interpret $(\chi _{1}(i))_{i=1}^{\infty }$ as a sequence and $\mathscr{A}$ as an infinite matrix $A$ with entries

$$ A_{ij} := \left\{ \textstyle\begin{array}{c} - |\chi _{0}(i) - \chi _{0}(j)|^{-a-2}, \quad \text{if } i \neq j \\ \sum_{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{\infty } |\chi _{0}(i) - \chi _{0}(k)|^{-a-2},\quad \text{if } i = j \end{array}\displaystyle \right\} , \quad \text{for } i,j \geq 1. $$

Since $A$ is strictly diagonally dominant and symmetric, $\zeta \mapsto \zeta ^{T} \mathscr{A} \zeta$ is a positive, strictly convex function on the space of sequences satisfying the matching condition $\zeta (i)- \zeta (i-1)\to 0$, and is thus uniquely globally minimised when $\zeta =0$. Since any $\chi _{1}$ satisfying $\chi _{1}(k) = O(k)$ and $\mathscr{A} \chi _{1} = 0$ also satisfies $\chi _{1}^{T} \mathscr{A} \chi _{1} = 0$, it follows that $\chi _{1} = 0$, which proves the claim.

A corollary of this claim is that any solution obtained to (74) subject to the matching condition $\chi _{1}(i) - \chi _{1}(i-1) \to p$ is unique, since otherwise the difference between two such solutions would be a nonzero solution to (74) that satisfies $\chi _{1}(i) - \chi _{1}(i-1) \to 0$.

From the form of (73), we observe that each higher correction $\chi _{j}$ will satisfy an infinite linear system of the form

$$ -a(a+1) \sum_{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{\infty } | \chi _{0}(i) - \chi _{0}(k)|^{-a-2} \bigl[ \chi _{j}(i) - \chi _{j}(k) \bigr] = g_{j}(i), \quad i = 1, 2, \ldots , $$

(75)

where $g_{j}(i)$ is obtained from the $\operatorname{ord}(n^{-\beta _{j}})$ terms in the multinomial expansion of $|\chi (i) - \chi (k)|^{-a-1}$, which in turn only depend on $\chi _{0}, \ldots , \chi _{j-1}$. Once an appropriate matching condition is specified in the form $\chi _{j}(i) - \chi _{j}(i-1) \to q_{j}$ for some constant $q_{j}$, we find that there will be a unique solution for $\chi _{j}$. Similarly, $\tilde{\chi }$ will satisfy a linear system of the form given in (75), and the identical style of matching condition will be required.

We note that $g_{j}(i)$ will only be nonzero if $\beta _{j}$ can be expressed as the sum of $\beta _{J}$ values (possibly including repetitions) where $J < j$. For example, $g_{2}$ will only be nonzero if $\beta _{2}$ is a multiple of $\beta _{1}$. Since the linear system for $\chi _{j}$ above is identical to the linear system for $\chi _{1}$, we see that $\chi _{j}(i) - \chi _{j}(i-1) \not \to 0$ as $i \to \infty$ is necessary for $\chi _{j}$ to have a nontrivial solution unless $g_{j}$ is nonzero. This is an important observation for performing the matched asymptotic analysis in Sect. A.4.

In the more general case where $V$ satisfies (30), we find with very minor modifications of the analysis above that each $\chi _{j}$ satisfies the infinite linear system

$$ -\sum_{{\scriptstyle k=0 \atop\scriptstyle k\neq i}}^{\infty } V'' \bigl[\chi _{0}(i) - \chi _{0}(k) \bigr] \bigl[ \chi _{j}(i) - \chi _{j}(k) \bigr] = g_{j}(i), \quad i = 1, 2, \ldots , $$

where the functions $g_{j}(i)$ are obtained from Taylor series expansions of $V'(\chi (i) - \chi (k))$.

1.4 A.4 Matching Between the Bulk and the Boundary Layer

We established in Sect. A.2 that $\xi$ can be expanded as an asymptotic series of the form (67) where $\xi _{k}(s) = (s - \frac{1}{2}) p_{k}$ and $\tilde{\xi }$ is given in (68). Additionally, we established in Sect. A.3 that $\chi$ can be expanded as an asymptotic series of the form (72), where $\chi _{j}$ for $j > 0$ and $\tilde{\chi }$ are all solutions to infinite linear systems subject to a condition of the form $\chi _{j}(i) - \chi _{j}(i-1) \to q_{j}$ (and similarly for $\tilde{\chi }$). However, we have not yet characterised the exponents $b_{k}$ and $\beta _{j}$ in the power series (67) and (72), neither have we determined the constants $p_{k}$, $\tilde{p}$, $q_{j}$ and $\tilde{q}$. We achieve this by using the method of matched asymptotic expansions.

We perform our asymptotic matching by introducing an intermediate matching variable, $R$, where $R$ is an integer with $1 \ll R \ll n$. We assert that this $R$ lies in the ‘overlap region’, so that both the bulk ansatz and the boundary layer ansatz yield asymptotic series solutions for $x(R;n)$ when $1 \ll R \ll n$. This involves making some assumptions about the asymptoticity of the bulk and boundary layer solutions outside the domains in which they are naturally defined. For example, we recall that we assumed that $i = \operatorname{ord}(n)$ in order to obtain the bulk equations described in Sect. A.1. We now assert that the bulk series solution obtained in Sect. A.2 remains valid whenever $i \gg 1$. That is, we assert that $\xi _{0}(Rn^{-1}) \gg n^{-b_{k}} \xi _{k}(Rn^{-1})$ for any $k > 0$ as long as $R \gg 1$. Despite the fact that $\tilde{\xi }(s)$ becomes unbounded as $s \to 0$, we observe that this assumption is consistent with comparing $\xi _{0}(s) = s$ with the solution for $\tilde{\xi }$ given in (68).

The matching variable, $R$, is distinct from the cut-off, $K$, used in several of the sums. We introduce the matching variable in order to analyse the relationship between the solution of the discrete boundary layer problem and the solution of the continuum bulk problem, whereas we introduce $K$ in order to account for the ‘bulk’ and ‘boundary layer’ contributions to the force on any individual particle.

Asymptotic matching requires that $\xi (Rn^{-1})$ and $n^{-1} \chi (R)$ should be asymptotically equivalent throughout the overlap region. That is, we require that

$$ \xi _{0} \bigl(Rn^{-1} \bigr) + n^{-b_{1}} \xi _{1} \bigl(Rn^{-1} \bigr) + \cdots \sim n^{-1} \chi _{0}(R) + n^{-\beta _{1}-1} \chi _{1}(R) + \cdots , $$

(76)

for all choices of $R$ with $1 \ll R \ll n$. Each term obtained from expanding $\xi (Rn^{-1})$ under the assumption that $Rn^{-1}$ is small should match with an equivalent term obtained from expanding $n^{-1} \chi (R)$ under the assumption that $R$ is large. In the case where logarithmic terms and related complications are absent, this can be conveniently expressed using a matching table, in which the rows represent asymptotic expansions of $n^{-b_{k}}\xi _{k}(Rn^{-1})$ for small $Rn^{-1}$ and the columns represent expansions of $n^{-\beta _{j} - 1} \chi _{j}(R)$ for large $R$. Every row and column of the matching table should be a valid asymptotic series when $1 \ll R \ll n$, and every term in the interior of the table should be asymptotically larger than the terms below and to the right.

In order to construct a plausible matching table, we begin by exploiting the information that we already have about the functions $\xi _{k}$ and $\tilde{\xi }$. Specifically, we observe from our analysis in Sect. A.2 that

$$ n^{-b_{k}}\xi _{k} \bigl(Rn^{-1} \bigr) = - \frac{p_{k}}{2} n^{-b_{k}} + p_{k} R n^{-b_{k} - 1}, \quad \text{wherever } b_{k} < a-1, $$

(77)

while the further assumption that $a \neq 2$ yields

$$\begin{aligned} n^{-(a-1)}\tilde{\xi } \bigl(Rn^{-1} \bigr) =& \frac{1 }{\zeta (a) (-a+2)_{4}}R^{-(a-2)}n^{-1} - \biggl( \frac{1}{\zeta (a) (-a+2)_{4}} + \frac{\tilde{p}}{2} \biggr) n^{-(a-1)} \\ & {}+ \biggl( \frac{1}{\zeta (a) (-a+1)_{3}} + \tilde{p} \biggr) Rn^{-a} + O \bigl(R^{2} n^{-(a+1)} \bigr). \end{aligned}$$

(78)

Based on these results, we construct the following ‘matching table’ where each row and column can be read as an equation:

$$\textstyle\begin{array}{@{}cccccccccccc} x(i;n) & \hspace{0.5em} \sim & \hspace{0.5em} n^{-1}\chi _{0}(R) & \hspace{0.5em} + & \hspace{0.5em} n^{-\beta _{1} - 1} \chi _{1}(R) & \hspace{0.5em} + & \hspace{0.5em} \cdots & \hspace{0.5em} + & \hspace{0.5em} n^{-a} \tilde{\chi }(R) & \hspace{0.5em} + & \hspace{0.4em} \cdots \\ \wr & \hspace{0.5em} & \hspace{0.5em} \wr & \hspace{0.5em} & \hspace{0.5em} \wr & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} \wr & \hspace{0.5em} & \hspace{0.4em} \\ \xi _{0}(Rn^{-1}) & \hspace{0.5em} = & \hspace{0.5em} Rn^{-1} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.4em} \\ + & \hspace{0.5em} & \hspace{0.5em} + & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.4em} \\ n^{-b_{1}}\xi _{1}(Rn^{-1}) & \hspace{0.5em} = & \hspace{0.5em} -\frac{p_{1}}{2}n^{-b_{1}} & \hspace{0.5em} + & \hspace{0.5em} p_{1} R n^{-b_{1} - 1} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.4em} \\ + & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} + & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.4em} \\ n^{-b_{2}}\xi _{2}(Rn^{-1}) & \hspace{0.5em} = & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} -\frac{p_{2}}{2}n^{-b_{2}} & \hspace{0.5em} + & \hspace{0.5em} \ldots & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.4em} \\ + & \hspace{0.5em} & \hspace{0.5em} + & \hspace{0.5em} & \hspace{0.5em} + & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.4em} \\ \vdots & \hspace{0.5em} & \hspace{0.5em} \vdots & \hspace{0.5em} & \hspace{0.5em} \vdots & \hspace{0.5em} & \hspace{0.5em} \ddots & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.4em} \\ + & \hspace{0.5em} & \hspace{0.5em} + & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.4em} \\ n^{-(a-1)} \tilde{\xi }(Rn^{-1}) & \hspace{0.5em} \sim & \hspace{0.5em} \tilde{\kappa }_{4} R^{-(a-2)}n^{-1} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} + & \hspace{0.5em} \cdots & \hspace{0.5em} + & \hspace{0.5em} (\tilde{\kappa }_{3} + \tilde{p}) R n^{-a} & \hspace{0.5em} + & \hspace{0.4em} \cdots \\ + & \hspace{0.5em} & \hspace{0.5em} + & \hspace{0.5em} & \hspace{0.5em} + & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} & \hspace{0.5em} + & \hspace{0.5em} & \hspace{0.4em} \\ \vdots & \hspace{0.5em} & \hspace{0.5em} \vdots & \hspace{0.5em} & \hspace{0.5em} \vdots & \hspace{0.5em} & \hspace{0.5em} \ddots & \hspace{0.5em} & \hspace{0.5em} \vdots & \hspace{0.5em} & \hspace{0.4em} \ddots \end{array} $$

where $\tilde{\kappa }_{r} := \frac{1 }{\zeta (a) (-a+2)_{r}}$.

The matching table illustrates the fact that each term in the expansions of $n^{-b_{k}}\xi _{k}(Rn^{-1})$ given in (77) and (78) must correspond to an equivalent term in the asymptotic expansion of one of the functions $n^{-\beta _{j}-1} \chi _{j}(R)$. While the entries in the matching table above are based on the expansions of $\xi _{k}$, the columns must also be valid series. This places significant restrictions on the choices of $b_{k}$ and $\beta _{j}$; for example, inspection of the $n^{-1} \chi _{0}(R)$ column of the matching table strongly suggests that $b_{1} = 1$.

More rigorously, we can determine the values of $b_{k}$ and $\beta _{j}$ without appealing directly to the matching table. Since $n^{-b_{k}}\xi _{k}(Rn^{-1})$ is given by (77) when $b_{k} < a-1$, we find that the only terms on the left hand side of (76) that take the form $c_{\tau \upsilon } n^{-\tau } R^{\upsilon }$ for nonzero $c_{\tau }\upsilon$ are terms where $\upsilon = 1$ or $\upsilon = 0$ or $\upsilon \leq \tau - a + 1$. This third possibility is associated with the case where $b_{k} \geq a-1$ and hence $n^{-b_{k}}\xi _{k}(Rn^{-1})$ may not be linear.

Since every term on the right hand side of (76) must balance with an identical term on the left hand side of (76), this implies that

$$ \chi _{j}(R) = \textstyle\begin{cases} q_{j} R + \hat{q}_{j} + O(R^{\beta _{j} - (a-2)} ), & \beta _{j} < a-2; \\ q_{j} R + O(R^{\beta _{j} - (a-2)} ), & a-2 \leq \beta _{j} < a-1; \end{cases} $$

(79)

where $q_{j}$ and $\hat{q}_{j}$ are constants. In order to match between equivalent terms on either side of (76), we find that the values of the constants $q_{j}$ and $\hat{q}_{j}$ will be associated with values of $p_{k}$. Since (79) is concerned with the behaviour of $\chi _{j}(R)$ when $R$ is large, we note that differencing (79) also provides justification of the fact that $\chi _{j}(i) - \chi _{j}(i-1)$ has a finite limit as $i \to \infty$ wherever $\beta _{j} < a-1$. More rigorously, this result could be established by exploiting the assumed differentiability of $\xi$ and considering asymptotic matching between $\xi '(Rn^{-1})$ and $\chi (R) - \chi (R-1)$.

Now, let us assume that there exists some $b_{k}$ in the range $0 < b_{k} < a-1$. By matching terms on either side of (76) and using (77), we find that both $b_{k}$ and $b_{k} - 1$ must be values taken by exponents $\beta _{j}$. Since the smallest $\beta _{j}$ is $\beta _{0} = 0$, it follows that $b_{k} \geq 1$. If $a < 2$, this leads to a contradiction with the requirement that $b_{k} < a-1$, and we would therefore conclude that there are no exponents $b_{k}$ in the range $0 < b_{k} < a-1$.

The case where $a > 2$ is a little more complicated. In order to analyse this problem, we recall from Sect. A.3 that if $0 < \beta _{j} < a-1$, then $q_{j} = 0$ implies either that $\chi _{j}(i)$ has only the trivial solution $\chi _{j}(i) \equiv 0$, or that $\beta _{j}$ can be expressed as the sum of other $\beta _{J}$ values (allowing possible repetitions), where all of these other $\beta _{J}$ are associated with nontrivial solutions for $\chi _{J}(i)$. Using this result, we show that the only possible $b_{k}$ with $0 < b_{k} < a-1$ are the integers.

For the purposes of contradiction, assume that $a > 2$ and that there exists some smallest noninteger $\theta$ in the range $0 < \theta < a-1$ so that $b_{k} = \theta$ is associated with a nontrivial solution for $\xi _{k}$. As noted above, this implies that both $\theta -1$ and $\theta$ must be values taken by the exponents $\beta _{j}$. Now, consider the function $\chi _{j}(i)$ associated with $\beta _{j} = \theta -1$. Since this has a nontrivial solution, it follows that either $q_{j} \neq 0$ or that $\theta - 1$ can be expressed as the sum of $\beta _{J}$ values associated with nontrivial solutions for $\chi _{J}$. However, $q_{j} \neq 0$ would imply that the matching table contains a term of the form $q_{j} R n^{-\theta }$, which must correspond to $\theta -1$ being a value taken by one of the $b_{k}$; this would be a contradiction with the assumption that $\theta$ is the smallest noninteger value of $b_{k}$. Similarly, if $\theta - 1$ can be expressed as a sum of $\beta _{J}$ values, at least one of these must be noninteger, which would also lead to a contradictory noninteger value of $b_{k}$ less than $\theta$.

For $a \neq 2$, we therefore find that the solution in the bulk region takes the form

$$ \xi (s) = s + \sum_{k=1}^{ \lceil a-2 \rceil } p_{k} \biggl( s - \frac{1}{2} \biggr) n^{-k} + n^{-(a-1)} \tilde{\xi }(s) + o \bigl(n^{-(a-1)} \bigr), $$

where $\tilde{\xi }(s)$ is given in (68). Using either intermediate matching (as described above) or Van Dyke’s matching criterion, we can use this expression to find the asymptotic behaviour of the functions $\chi _{j}(i)$ as $i \to \infty$. It follows that the solution in the boundary layer region takes the form

$$ \chi (i) = \chi _{0}(i) + \sum _{j=1}^{ \lceil a-2 \rceil } \chi _{j}(i) n^{-j} + n^{-(a-1)} \tilde{\chi }(i) + o \bigl(n^{-(a-1)} \bigr). $$

(80)

Moreover, we can use the matching table to define the asymptotic behaviour of $\chi _{j}(i)$ as $i \to \infty$ in terms of the constants $p_{k}$ and $\tilde{p}$ from the solution in the bulk region. Specifically, we find that the asymptotic behaviour of $\chi _{j}(i)$ for large $i$ and $a > 2$ is given by

$$ \chi _{j}(i) = \textstyle\begin{cases} i - \frac{p_{1}}{2} + \frac{1 }{\zeta (a) (-a+2)_{4}}i^{-(a-2)} + o\bigl(i^{-(a-2)}\bigr), & j = 0; \\ p_{j} i - \frac{p_{j+1}}{2} + O \bigl(i^{-(a-2-j)} \bigr), & 0 < j < a-2; \\ p_{j} i - \biggl( \frac{1}{\zeta (a) (-a+2)_{4}} + \frac{\tilde{p}}{2} \biggr) + o(1), & j = a-2; \\ p_{j} i + O \bigl(i^{-(a-2-j)} \bigr), & a-2 < j < a-1. \end{cases} $$

(81)

The behaviour of $\tilde{\chi }(i)$ for large $i$ is given by

$$ \tilde{\chi }(i) = \biggl( \frac{1}{\zeta (a) (-a+1)_{3}} + \tilde{p} \biggr) i + o(i). $$

These expressions enable us to define the constants $p_{j}$ based on the solutions obtained for $\chi _{j}(i)$. For $1 \leq j < a-1$, we see that

$$ p_{j} = 2\lim_{i \to \infty } \bigl[ p_{j-1} i - \chi _{j-1}(i) \bigr] , $$

where we take $p_{0} = 1$. If $a$ is an integer, we also find that

$$ \tilde{p} = 2\lim_{i \rightarrow \infty } \bigl[ p_{a-1} i - \chi _{a-1}(i) \bigr] - \frac{2}{\zeta (a) (-a+2)_{4}}. $$

If $a$ is not an integer, we require that $\tilde{p} = -\frac{2}{\zeta (a) (-a+2)_{4}}$. If this were not the case, (78) would yield an $\operatorname{ord}[R^{0} n^{-(a-1)}]$ term on the left hand side of (76) that could not be balanced by any equivalent term on the right hand side of (76) without contradicting the result that $\chi$ has an expansion of the form given in (80).

In the case where $a < 2$, we recall that $\xi (s)$ must take the form

$$ \xi (s) = s + n^{-(a-1)} \biggl[ \frac{s^{-(a-2)} - (1-s)^{-(a-2)}}{\zeta (a) (-a+2)_{4}} + \biggl(s - \frac{1}{2} \biggr)\tilde{p} \biggr] + o \bigl(n^{-(a-1)} \bigr). $$

By the same argument as above for noninteger $a$ when $a > 2$, we find that $\tilde{p} = -\frac{2}{\zeta (a) (-a+2)_{4}}$ also when $a < 2$. Hence, we find from (80) that $\chi (i) = \chi _{0}(i) + n^{-(a-1)}\tilde{\chi }(i)$ and that the asymptotic behaviours of these functions are given by

$$ \chi _{0}(i) \sim i + \frac{1 }{\zeta (a) (-a+2)_{4}}i^{-(a-2)} + o \bigl(i^{-(a-2)} \bigr), $$

(82)

and

$$ \tilde{\chi }(i) = \frac{-a}{\zeta (a)(-a+2)_{4}} i + o(i). $$

In the case where $a = 2$, the logarithm in (68) requires careful handling, and we find that some additional terms that are logarithmically large in $n$ need to be introduced. This makes it more difficult to construct a matching table, but the arguments described above can still be used with some modifications. Ultimately, we find that we can account for all logarithmic terms using the expansions

$$ \xi (s) = s + n^{-1} (\log n) \biggl(s - \frac{1}{2} \biggr) \tilde{p}^{*} + n^{-1} \biggl[ \frac{\log (1-s) - \log (s) }{\pi ^{2}} + \biggl(s - \frac{1}{2} \biggr)\tilde{p} \biggr] + o \bigl(n^{-1} \bigr) $$

and

$$ \chi (i) = \chi _{0}(i) + n^{-1} (\log n) \tilde{\chi }^{*}(i) + n^{-1} \tilde{\chi }(i) + o \bigl(n^{-1} \bigr). $$

(83)

Matching between the bulk and the boundary layer can then be achieved by setting $\tilde{p}^{*} = \frac{2}{\pi ^{2}}$, and taking

$$ \tilde{p} = 2 \lim_{i \rightarrow \infty } \bigl[ i - \chi _{0}(i) \bigr] . $$

While we have concentrated on obtaining terms up to $\operatorname{ord}[n^{-(a-1)}]$ in our expansions of both $\xi$ and $\chi$, it may be noted that further high order terms can also be obtained using the techniques of matched asymptotic expansions. However, obtaining these high-order terms becomes much more algebraically laborious. In Sect. A.1, we commented that finding higher-order corrections requires us to expand $S_{0}$ in (55) and exploiting the properties of $\chi (i)$. In the same way, obtaining higher order corrections in the boundary layer would require us to expand $S_{5}$ and $S_{6}$ in (70) and exploit the properties of $\xi (s)$. Additionally, we find that the high order solutions for $\xi _{k}$ are no longer as simple as the expressions obtained when $b_{k} < a-1$, which causes the matching table to become much more complicated.

As described in this section, formal asymptotic methods can be used to elucidate the structure of the original discrete problem and determine the appropriate scalings for higher-order asymptotic analysis. By the principles of matched asymptotic expansions, we use information about the behaviour of the bulk solution to construct the boundary layer solution and vice versa; this is where formal asymptotic analysis becomes particularly useful. For example, our higher-order analysis of $\xi$ gives us detailed information about the decay properties of $\chi _{0}$. Indeed, combining (81), (82), and (83), we obtain the decay properties of $\chi _{0}$ as given by (28), from which (4) follows.

In the general case where $V$ satisfies (30), the coefficients of various terms change but the structure of the asymptotic matching remains identical up to $\operatorname{ord}(n^{-(a-1)})$. Hence, we also find that the solution for $\tilde{\xi }$ given in (69) can be used to obtain information about the decay behaviour of $\chi _{0}$ for a general $V$. From this, we find that we can generalise (4) to obtain (31).

Appendix B: Technical Details and Computations of Sect. 4

Given the notation in Sect. 4, we prove inequalities (40a) and (40b) in Sect. B.1, and derive the second equality in (34) in Sect. B.2.

2.1 B.1 Proof of the Inequalities (40a) and (40b) of Theorem 4.3

Proof of (40a)

Let $\epsilon ^{n} \stackrel{ 2 }{\rightharpoonup } (\epsilon , \hat{\epsilon })$ such that $E_{n}^{1} (\epsilon ^{n})$ is bounded. For the second term of $E_{n}^{1} (\epsilon ^{n})$ in (34), we use the strong convergence of $\sigma ^{n}$ (see Lemma 4.2) to obtain

$$ \bigl( \sigma ^{n}, \epsilon ^{n} + \hat{ \epsilon }^{n} \bigr)_{\ell ^{2}(\mathbb{N})} \xrightarrow{ n \to \infty } \bigl( \sigma ^{\infty }, \epsilon + \hat{\epsilon } \bigr)_{\ell ^{2}(\mathbb{N})}. $$

(84)

We bound $Q_{n}(\epsilon _{n})$ in (34) from below by dropping some terms in the summation. We set $R_{n} := \lfloor n/2 \rfloor$, and estimate

$$ Q_{n}(\epsilon _{n}) = \sum_{k=1}^{n} \sum_{j=0}^{n-k} \phi _{k} \Biggl( \sum_{l = j+1}^{k+j} \epsilon ^{n}(l) \Biggr) \geq \sum_{k=1}^{R_{n}} \sum _{j=0}^{R_{n} - k} \Biggl[ \phi _{k} \Biggl(\sum _{l = j+1}^{k+j} \epsilon ^{n}(l) \Biggr) + \phi _{k} \Biggl(\sum_{l=j+1}^{k+j} \hat{\epsilon }^{n}(l) \Biggr) \Biggr]. $$

To pass to the liminf as $n \to \infty$, we use Fatou’s Lemma, by which we interpret the double sum as an integral over the lattice $\mathbb{N}_{+}^{ 2 }$. We focus on the first term in the summand, because the second term involving $\hat{\epsilon }^{n}$ can be estimated analogously. For the pointwise lower bound (as $n \to \infty$ with $k$ and $j$ fixed) of the summand, we interpret $\sum_{l = j+1}^{k+j} \epsilon ^{n}_{l}$ as an inner product of $\epsilon ^{n}$ with an $n$-independent sequence consisting of 1’s and 0’s. Then the fact that $\epsilon ^{n, 1/2} \rightharpoonup \epsilon$ implies

$$ \sum_{l = j+1}^{k+j} \epsilon ^{n, 1/2}(l) \xrightarrow{n \to \infty } \sum_{l = j+1}^{k+j} \epsilon (l) \quad \text{for all } k, j \geq 1. $$

Since Lemma 4.1 implies that $\phi _{k}$ is positive and lower semicontinuous, it follows that

$$ \liminf_{n\to \infty } \phi _{k} \Biggl(\sum _{l = j+1}^{k+j}\epsilon ^{n, 1/2}_{l} \Biggr) \geq \phi _{k} \Biggl(\sum_{l = j+1}^{k+j} \epsilon (l) \Biggr) \quad \text{for all } k, j \geq 1. $$

This shows that the hypotheses of Fatou’s Lemma are satisfied, and thus

$$ \liminf_{n\to \infty } \sum_{k=1}^{n} \sum_{j=0}^{n-k} \phi _{k} \Biggl( \sum_{l = j+1}^{k+j} \epsilon ^{n}(l) \Biggr) \geq \sum_{k=1}^{\infty }\sum _{j=0}^{\infty } \Biggl[ \phi _{k} \Biggl(\sum _{l = j+1}^{k+j} \epsilon (l) \Biggr) + \phi _{k} \Biggl(\sum_{l=j+1}^{k+j} \hat{\epsilon }(l) \Biggr) \Biggr]. $$

 □

Proof of (40b)

Let $(\epsilon , \hat{\epsilon }) \in \operatorname{Dom}E_{\infty }^{1}$ such that $E_{\infty }^{1} (\epsilon , \hat{\epsilon }) =: C < \infty$. Then $\epsilon , \hat{\epsilon }\in \ell ^{2} (\mathbb{N}) \subset \ell ^{\infty }(\mathbb{N})$, and

$$ E^{1}_{\infty }(\epsilon , \hat{\epsilon }) \geq \phi _{1} \bigl( \epsilon (i) \bigr) + \phi _{1} \bigl( \hat{ \epsilon }(j) \bigr) - \| \sigma ^{\infty } \|_{ \ell ^{2}(\mathbb{N}) } \| \epsilon + \hat{\epsilon } \|_{ \ell ^{2}(\mathbb{N}) } \quad \text{for all } i,j \geq 1. $$

Hence, there exists a $\delta > 0$ such that

$$ \epsilon , \hat{\epsilon }\in X_{\delta }:= \biggl\{ \epsilon \in \ell ^{2} (\mathbb{N} ) \mid \delta - 1 \leq \epsilon (i) \leq \frac{1}{\delta } ,\text{ for all }i\in \mathbb{N} \biggr\} . $$

Next we construct a recovery sequence. As in [23], we note that the constraint that $\sum_{i=1}^{\infty }\epsilon ^{n}(i) = 0$ need not be preserved in the limit as $n\to \infty$. We take this into account by introducing $1 \ll S_{n} \ll n$ as the index where we match the boundary layer with the bulk. We note that as $n\to \infty$, it holds that $S_{n}\to \infty$ and $S_{n}/n\to 0$. We further set

$$ u_{n} := \sum_{i=1}^{S_{n}} \epsilon (i), \quad \text{and} \quad \hat{u}_{n} := \sum _{i=1}^{S_{n}} \hat{\epsilon }(i). $$

(85)

We now define the recovery sequence

$$ \epsilon ^{n}(i) := \textstyle\begin{cases} {\epsilon }(i) & i\in \{1,\ldots ,S_{n}\},\\ -\dfrac{u_{n} + \hat{u}_{n}}{n-2S_{n}} &i\in \{S_{n}+1,\ldots ,n-S_{n}\},\\ \hat{\epsilon }(n+1-i) & i\in \{n-S_{n}+1,\ldots ,n\}. \end{cases} $$

(86)

It is easily checked that $\sum_{i=1}^{n} \epsilon ^{n}(i) = 0$ and $\epsilon ^{n}(i) \geq -1$ for $n$ large enough, hence $\epsilon ^{n} \in \operatorname{Dom}E_{n}^{1}$.

To show that $\epsilon ^{n} \xrightarrow{ 2 } (\epsilon , \hat{\epsilon })$, we prove that $\epsilon ^{n,1/2} \to \epsilon$ in $\ell ^{2} (\mathbb{N})$, and conclude by an analogous argument that $\hat{\epsilon }^{n,1/2} \to \hat{\epsilon }$ in $\ell ^{2} (\mathbb{N})$. To this end, we estimate

$$\begin{aligned} \| \epsilon - \epsilon ^{n,1/2} \|_{ \ell ^{2} (\mathbb{N}) } ^{2} =& \sum_{i=S_{n}+1}^{ \lceil n/2 \rceil } \biggl( \epsilon (i) + \frac{u_{n} + \hat{u}_{n}}{n-2S_{n}} \biggr)^{2} + \sum_{i=\lceil n/2 \rceil + 1}^{\infty } \epsilon (i)^{2} \\ \leq& 2 \sum_{i=S_{n}+1}^{ \lceil n/2 \rceil } \biggl( \frac{u_{n} + \hat{u}_{n}}{n-2S_{n}} \biggr)^{2} + 2 \sum _{i=S_{n}+1}^{\infty } \epsilon (i)^{2}. \end{aligned}$$

(87)

The second term in the right-hand side of (87) converges to 0 as $n \to \infty$ because $\epsilon \in \ell ^{2} (\mathbb{N})$. To show that the first term in the right-hand side is also small for large $n$, we interpret $u_{n}$ in (85) as the inner product of $\epsilon$ with a sequence consisting of 1’s and 0’s. Applying the Cauchy-Schwartz inequality on this inner product yields

$$ \sum_{i=S_{n}+1}^{ \lceil n/2 \rceil } \biggl( \frac{u_{n} + \hat{u}_{n}}{n-2S_{n}} \biggr)^{2} \leq \frac{1}{(n-2S_{n})^{2}} \sum _{i=S_{n}+1}^{ \lceil n/2 \rceil } ( 2 \sqrt{S_{n}} \| \epsilon \|_{ \ell ^{2} (\mathbb{N}) } )^{2} \lesssim \frac{1}{n^{2}} \frac{n}{2} 4 S_{n} \| \epsilon \|_{ \ell ^{2} (\mathbb{N}) } ^{2} \xrightarrow{ n \to \infty } 0, $$

where we recall that $S_{n}\ll n$ as $n\to \infty$. This completes the proof of $\epsilon ^{n} \xrightarrow{ 2 } (\epsilon , \hat{\epsilon })$.

To establish the limsup inequality (40b), we observe from the argument leading to (84) that it is enough to focus on $Q_{n}$ in (34), since the convergence of terms involving $\sigma ^{n}$ is implied by the fact that $\epsilon ^{n} \xrightarrow{ 2 } (\epsilon , \hat{\epsilon })$, as just shown. For convenience, we choose $S_{n}$ to be even. We split the summation in $Q_{n}$ into four parts:

$$\begin{aligned} Q_{n} \bigl(\epsilon ^{n} \bigr) =& \sum _{k=1}^{S_{n}/2} \Biggl[ \sum _{j=0}^{S_{n}-k} \phi _{k} \Biggl(\sum _{l = j+1}^{k+j} \epsilon ^{n}(l) \Biggr) + \sum_{j=S_{n}-k + 1}^{n - S_{n} - 1} \phi _{k} \Biggl(\sum_{l = j+1}^{k+j} \epsilon ^{n}(l) \Biggr) \\ & {}+ \sum_{j = n-S_{n}}^{n-k} \phi _{k} \Biggl(\sum_{l = j+1}^{k+j} \epsilon ^{n}(l) \Biggr) \Biggr] + \sum_{k = 1 + S_{n}/2}^{n} \sum_{j=0}^{n-k} \phi _{k} \Biggl( \sum_{l = j+1}^{k+j} \epsilon ^{n}(l) \Biggr). \end{aligned}$$

(88)

The first and third term are constructed to contain only those elements of $\epsilon ^{n}(l)$ which equal either $\epsilon (l)$ or $\hat{\epsilon }(l)$. Using this observation, we estimate these terms by

$$\begin{aligned} &\sum_{k=1}^{S_{n}/2} \Biggl[ \sum _{j=0}^{S_{n}-k} \phi _{k} \Biggl(\sum _{l = j+1}^{k+j} \epsilon ^{n}(l) \Biggr) + \sum_{j = n-S_{n}}^{n-k} \phi _{k} \Biggl(\sum_{l = j+1}^{k+j} \epsilon ^{n}(l) \Biggr) \Biggr] \\ &\quad \leq \sum_{k=1}^{\infty }\sum _{j=0}^{\infty } \Biggl[ \phi _{k} \Biggl(\sum _{l = j+1}^{k+j} \epsilon (l) \Biggr) + \phi _{k} \Biggl(\sum_{l=j+1}^{k+j} \hat{\epsilon }(l) \Biggr) \Biggr]. \end{aligned}$$

Since the right-hand side equals the first two terms of $E_{\infty }^{1}$ given by $Q_{\infty }$ and $\hat{Q}_{\infty }$, it remains to show that the second and fourth term in (88) converge to 0 as $n \to \infty$.

We start by proving that the second term is small for large $n$. We observe that it solely contains those elements of $\epsilon ^{n}(i)$ which equal either entries of the tails of $\epsilon$ and $\hat{\epsilon }$, or equal the (small) constant term in (86). For this reason, it turns out to be enough to bound the second term by employing the quadratic upper bound of $\phi _{k}$ given by Lemma 4.1 (it applies because of $\epsilon ^{n} \in X_{\delta }$), and then applying Jensen’s inequality. In more detail

$$\begin{aligned} & \sum_{k=1}^{S_{n}/2} \sum _{j=S_{n}-k + 1}^{n - S_{n} - 1} \phi _{k} \Biggl(\sum _{l = j+1}^{k+j} \epsilon ^{n}(l) \Biggr) \\ &\quad \leq \sum_{k=1}^{S_{n}/2} \sum _{j=S_{n}-k + 1}^{n - S_{n} - 1} \frac{C_{\delta }}{k^{a+2}} k^{2} \Biggl(\frac{1}{k} \sum_{l = j+1}^{k+j} \epsilon ^{n}(l) \Biggr)^{2} \\ &\quad \leq \sum_{k=1}^{S_{n}/2} \frac{C_{\delta }}{k^{a+1}} \sum_{j=S_{n}-k + 1}^{n - S_{n} - 1} \sum _{l = j+1}^{k+j} | \epsilon ^{n}(l) |^{2} \leq \sum_{k=1}^{S_{n}/2} \frac{C_{\delta }}{k^{a+1}} \sum_{i= S_{n}/2 + 2}^{n - S_{n}/2 - 1} k | \epsilon ^{n}(i) |^{2} \\ &\quad \leq \Biggl(\sum_{k=1}^{\infty } \frac{C_{\delta }}{k^{a}} \Biggr) \Biggl[ \sum_{i=S_{n}/2}^{\infty } \bigl( |\epsilon (i)|^{2} + |\hat{\epsilon }(i)|^{2} \bigr) + \sum_{i=S_{n}+1}^{n - S_{n}} \biggl( \frac{u_{n} + \hat{u}_{n}}{n-2S_{n}} \biggr)^{2} |\epsilon ^{n}(i)|^{2} \Biggr], \end{aligned}$$

in which the right-hand side converges to 0 as $n\to \infty$ by the same argument that we use for showing the convergence of the right-hand side in (87).

Finally, we show that the fourth term in (88) converges to 0. Since $k$ (the distance between particles in terms of their index) is large, we use similar arguments in the following estimate

$$\begin{aligned} &\sum_{k = 1 + S_{n}/2}^{n} \sum _{j=0}^{n-k} \phi _{k} \Biggl(\sum _{l = j+1}^{k+j} \epsilon ^{n}(l) \Biggr) \\ &\quad \leq \sum_{k = 1 + S_{n}/2}^{n} \sum _{j=0}^{n-k} \frac{C_{\delta }}{k^{a+2}} k^{2} \Biggl(\frac{1}{k} \sum_{l = j+1}^{k+j} \epsilon ^{n}(l) \Biggr)^{2} \\ &\quad \leq \sum_{k = 1 + S_{n}/2}^{n} \frac{C_{\delta }}{k^{a+1}} \sum_{j=0}^{n-k}\sum _{l = j+1}^{k+j} | \epsilon ^{n}_{l} |^{2} \leq \sum_{k = 1 + S_{n}/2}^{n} \frac{C_{\delta }}{k^{a}} \sum_{i=1}^{n} | \epsilon ^{n}(i) |^{2} = C_{\delta } \| \epsilon ^{n} \|_{ \ell ^{2} (\mathbb{N}) } ^{2} \sum _{k = 1 + S_{n}/2}^{n} \frac{1}{k^{a}}, \end{aligned}$$

which converges to 0 as $n \to \infty$ since $a > 3/2 > 1$. □

2.2 B.2 Computation of $\sigma ^{n}$

In this section we fix $n$, and show that

$$ \sum_{k=1}^{n} \sum _{j=0}^{n-k} \sum _{l=j+1}^{k+j} V'(k) \epsilon (l) = \sigma \cdot (\epsilon + \hat{\epsilon }), $$

(89)

where $\sigma$ is given by (35), i.e.

$$ \sigma _{i} = \sum_{k = i+1}^{n-i} \bigl[ (k-i) \wedge (n-i+1-k) \bigr] | V' (k) | \quad \text{for } i = \bigl\{ 1, \ldots , \lfloor n/2 \rfloor \bigr\} . $$

We start by changing the order of summation in

$$\begin{aligned} \sum_{k=1}^{n} V'(k) \sum _{j=0}^{n-k} \sum _{l=j+1}^{k+j} \epsilon (l) =& \sum _{k=1}^{n} \sum_{l=1}^{n} \epsilon (l) V'(k) \sum_{j = 0 \vee (l - k) }^{ (n-k) \wedge (l-1) } 1 \\ =& \sum_{k=1}^{n} \sum _{l=1}^{n} V'(k) \bigl[ k \wedge l \wedge (n-k+1) \wedge (n-l+1) \bigr] \epsilon (l) =: v \cdot \tilde{A} \epsilon , \end{aligned}$$

where the vector $v \in \mathbb{R}^{ n }$ is defined by $v_{k} := V'(k) < 0$, and the matrix $\tilde{A} \in \mathbb{R}^{ n \times n }$ is illustrated in Fig. 13.

Fig. 13

Schematic picture of the two index matrices $\tilde{A}$ and $A = \tilde{A} - B$. The lines are level sets which connect entries with the same value. We have taken $n$ to be even for convenience

Since $\epsilon \cdot \mathbf{1} = 0$, it holds that $\tilde{A} \epsilon = (\tilde{A} - B) \epsilon$ for any matrix $B$ whose rows are multiples of $\mathbf{1}$. We take $B$ such that the entries in its $i$-th row equal $i \wedge (n-i+1)$, and set $A := -(\tilde{A} - B)^{T}$ (see Fig. 13 for its structure). Then

$$ v \cdot \tilde{A} \epsilon = -v \cdot A^{T} \epsilon = -\epsilon \cdot A v = \sum_{l = 1}^{\lfloor n/2 \rfloor } \bigl[ (-A v) (l) \epsilon (l) + (-A v) (n-l+1) \hat{\epsilon }(l) \bigr]. $$

From Fig. 13 it is easy to see that the vector $-A v$ has reversal symmetry, and (89) follows.