# Exact finite-size corrections in the dimer model on a cylinder.

Vladimir V. Papoyan<EMAIL_ADDRESS>Bogoliubov Laboratory of Theoretical
Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russian
Federation  Dubna State University, Dubna, Russian Federation

###### Abstract

The exact finite-size corrections to the free energy $F$ of the dimer model on
lattice $\mathcal{M}\times\mathcal{N}$ with cylindrical boundary conditions
have been derived for three cases where the lattice is completely covered by
dimers: $\mathcal{M}=2M$, $\mathcal{N}=2N$; $\mathcal{M}=2M-1$,
$\mathcal{N}=2N$; and $\mathcal{M}=2M$, $\mathcal{N}=2N-1$. For these types of
cylinders, ratios $r_{p}(\rho)$ of the $p$th coefficient of $F$ have been
calculated for the infinitely long cylinder (${\mathcal{M}}\rightarrow\infty$)
and infinitely long strip (${\mathcal{N}}\rightarrow\infty$) at varying aspect
ratios. As in previous studies of the dimer model on the rectangular lattice
with free boundary conditions and for the Ising model with Brascamp-Kunz
boundary conditions, the limiting values $p\to\infty$ exhibit abrupt anomalous
behaviour of ratios $r_{p}(\rho)$ at certain values of $\rho$. These critical
values of $\rho$ and the limiting values of the finite-size expansion
coefficient ratios vary between the different models.

###### pacs:

05.50+q, 05.70.Fh, 75.10-b

##  I Introduction

Finite-size scaling, introduced by Fisher and Barber in 1972, is an essential
tool for studying the thermodynamic properties of systems in any dimension
FisherBarber . It allows us to extract critical and non-critical properties
from finite-size systems, which can be extrapolated then to infinite systems.
Finite systems have boundaries, and these boundaries must be taken into
account even if they don’t play a role in the thermodynamic limit.
Understanding the role of boundary conditions is crucial for the understanding
of critical phenomena in these systems.

Very few models have analytical or exact solutions. Two prime examples of
models that do have these solutions are the Ising model and the dimer model in
low dimensions. These analytical results are important for understanding of
the correct forms of finite-size scaling in these models, as they help us to
avoid the stochastic errors that can occur in simulation methods. Therefore,
these results are essential for gaining a better understanding of the role of
boundary conditions in these models

Izmailian, Papoyan and Ziff Izmailian2019  considered recently the dimer model
on the rectangular $2M\times 2N$ lattice with free boundary conditions. In
doing so, they built on work by Kasteleyn Kasteleyn , as well as by Temperley
and Fisher FisherTemp , who in the 1960’s found the first few coefficients of
the asymptotic expansion of the free energy. This is still considered as one
of the most important exact results of fundamental statistical mechanics.
Izmailian, Papoyan and Ziff derived exact expressions for these coefficients
in terms of the elliptic theta functions $\theta_{2},\theta_{3},\theta_{4}$
and the elliptic integral of second kind $E$, up to 22nd order.

One of the main findings from this study was very intriguing result that the
asymptotic coefficients for the square and infinitely long strip, for large
orders, approach exactly a factor of 1/2 in ratio. While the ratio between the
asymptotic coefficients for rectangles and infinitely long strips approaches
exactly 1 as the order increases, when scaled properly by the aspect ratio.
This allows better understanding of the coefficients in the asymptotic
behavior of the square and rectangle, for example, because of the simpler form
for the coefficients of the strip.

To understanding in depth this abrupt anomalous behaviour, the authors suggest
considering other available models. One such model is the Ising model with
Brascamp-Kunz boundary conditions, which has been studied by Izmailian, Kenna
and Papoyan in Ref.Izmailian2023 . They derived exact finite-size corrections
for the free energy $F$ of the Ising model on the ${\cal M}\times 2{\cal N}$
square lattice with Brascamp-Kunz boundary conditions. The ratios
$r_{p}(\rho)$ of $p$th coefficients of $F$ for the square lattice to
coefficients of $F$ for the infinitely long cylinder (${\cal M}\to\infty$) and
to coefficients of $F$ for the infinitely long Brascamp-Kunz strip (${\cal
N}\to\infty$) at varying values of the aspect ratio $\rho={(\cal M}+1)/2{\cal
N}$ were calculated as well. Similarly to previous studies for the dimer model
on a rectangular $2M\times 2N$ lattice with free boundary conditions, the
limiting values of $p\rightarrow\infty$ for $r_{p}(\rho)$ exhibit abrupt
anomalous behavior at certain values of $\rho$. For increasing $p$, the values
of the ratios $r_{p}(\rho)$ tend to 1 for $\rho\neq 1/2$, while they vanish
exactly at $\rho=1/2$ in both infinitely long cylinder and infinitely long
Brascamp-Kunz strip cases. However, critical values of $\rho$ and the limiting
values of the finite-size-expansion coefficient ratios differ, between dimer
model on the rectangular and Ising model with Brascamp–Kunz boundary
conditions.

The aim of the present paper is to gain a better understanding of the abrupt
change in the behavior of the $r_{p}(\rho)$ ratio by examining it in the dimer
model with cylindrical boundary conditions.

##  II Dimer model on cylinder lattice

The partition function of the dimer model on $\mathcal{M}\times\mathcal{N}$
lattice is given by

$Z_{\mathcal{M},{\mathcal{N}}}(z_{v},z_{h})=\sum z_{v}^{n_{v}}z_{h}^{n_{h}}$
(1)

where the summation is taken over all possible dimer covering configurations,
$z_{h}$ and $z_{v}$ are the dimer weight in the horizontal and vertical
directions respectively, and $n_{v}$ and $n_{h}$ are the number of vertical
and horizontal dimers respectively Kasteleyn .

The explicit expression of the partition function depends crucially on whether
$\mathcal{M}$ and $\mathcal{N}$ are even or odd, and since the total number of
sites must be even if the lattice is to be completely covered by dimers, we
will consider three cases $\mathcal{M}=2M$, $\mathcal{N}=2N$;
$\mathcal{M}=2M-1$, $\mathcal{N}=2N$; $\mathcal{M}=2M$, $\mathcal{N}=2N-1$.

In accordance with results for isotropic case ($z_{v}=z_{h}=z=1$) from Ref.
Izmailian2003 , Eq. (1) can be written in terms of the partition function with
twisted boundary conditions $Z_{\alpha,\beta}(M,N)$ which is defined as
follows:

$\displaystyle Z_{\alpha,\beta}^{2}(M,N)$ $\displaystyle=$
$\displaystyle\prod_{n=0}^{N-1}\prod_{m=0}^{M-1}4\left[\sin^{2}\left(\frac{\pi(n+\alpha)}{N}\right)+\sin^{2}\left(\frac{\pi(m+\beta)}{M}\right)\right].$
(2)

Then, for the three cases of dimers under consideration, the results of the
partition function on cylinder can be reduced to:

$\displaystyle Z_{2M,2N}^{\mathrm{cyl}}(1)$ $\displaystyle=$
$\displaystyle\frac{Z_{\frac{1}{2},\frac{1}{2}}(2M+1,N)}{2\cosh{\left(N{\rm
arcsinh}1\right)}}$ (3)

for dimers on $2M\times 2N$ cylinder,

$\displaystyle Z_{2M-1,2N}^{\mathrm{cyl}}(1)$ $\displaystyle=$
$\displaystyle\frac{Z_{\frac{1}{2},0}(2M,N)}{2\cosh{\left(N{\rm
arcsinh}1\right)}}$ (4)

for dimers on $(2M-1)\times 2N$ cylinder,

$\displaystyle Z_{2M,2N-1}^{\mathrm{cyl}}(1)$ $\displaystyle=$
$\displaystyle\left[\frac{Z_{0,\frac{1}{2}}(2M+1,2N-1)}{2\cosh{\left((2N-1){\rm
arcsinh}1\right)}}\right]^{\frac{1}{2}}$ (5)

for dimers on $2M\times(2N-1)$ cylinder.

##  III Finite-size scaling theory

Ivashkevich, Izmailian, and Hu Izmailian2002  proposed a systematic method for
computing finite-size corrections to the partition function for free models on
a torus. They focused on the dimer model on $2M\times 2N$ lattice, as well as
the Ising and Gaussian models on $M\times N$ lattice. The authors derived all
terms in the exact asymptotic expansion of the logarithm of the partition
function for a class of free, exactly solvable statistical mechanics models.
Their approach is based on the intimate relationship between the terms of
asymptotic expansions and Kronecker’s double series. This work has been
further developed by Izmailian, Oganesyan, and Hu Izmailian2003  for the dimer
model on a rectangular $M\times N$ lattice with various boundary conditions
and different parities of lattice sites in the vertical $M$ and horizontal $N$
directions. Later, Izmailian et al. in Izmailian2014  derived the exact
asymptotic expansions of the partition functions for the dimer model on a
rectangular $(2M-1)\times(2N-1)$ lattice with a single monomer located on the
boundary, under both free and cylindrical boundary conditions. In particular
they show that because of certain non-local features present in the model, the
finite-size corrections in a crucial way depend on the parity of the lattice
sites in horizontal $N$ and vertical $M$ directions. Furthermore, the change
of parity of $M$ or $N$ induces a change of boundary condition Izmailian2005 ;
Izmailian2007 .

It has been shown Izmailian2003 ; Izmailian2019  that the exact asymptotic
expansion of the free energy for dimers on an open rectangular
$\mathcal{M}\times\mathcal{N}$ lattice takes the following form

$f=f_{\text{bulk}}+\frac{f_{1s}}{\mathcal{M}}+\frac{f_{2s}}{\mathcal{N}}+f_{\text{corn}}\frac{\ln
S}{S}+\frac{f_{0}}{S}+\sum_{p=1}^{\infty}\frac{f_{p}}{S^{p+1}}.$ (6)

Here $S=\mathcal{M}\times\mathcal{N}$ is the area of the lattice. The bulk
free energy $f_{\text{bulk}}$ is same for all boundary conditions
Izmailian2003 :

$f_{\mathrm{bulk}}=\frac{1}{2\pi}\int_{0}^{\pi}\omega_{1}(x)dx=\frac{1}{2\pi}\int_{0}^{\pi}{\rm
arcsinh}{\left(\sin{x}\right)}dx=\frac{G}{\pi},$ (7)

where $G$ is the Catalan constant. For the cylindric boundary conditions the
surface free energies are $f_{1s}=-\frac{1}{2}\ln{(1+\sqrt{2})}=-0.78726...$,
$f_{2s}=0$ and corner free energies $f_{\text{corn}}=0$ Izmailian2003 . The
leading finite-size correction term is $f_{0}$ and the subleading correction
terms are $f_{p}$ for $p=1,2,3,...$. Indeed such a form of the asymptotic
expansion for the free energy also holds for the Ising model, the spanning-
tree model, the Gaussian model, and resistor networks.

The exact asymptotic expansion of the logarithm of the partition function
according the Izmailian2002 ; Izmailian2003  can be written as

$\displaystyle\ln Z_{\alpha,\beta}(z,\mathcal{M},\mathcal{N})$
$\displaystyle=$
$\displaystyle\frac{\mathcal{S}}{\pi}\int_{0}^{\pi}\\!\\!\omega_{z}(x)~{}\\!{\rm
d}x+\ln\frac{\theta_{\alpha,\beta}}{\eta}-2\pi\xi\sum_{p=1}^{\infty}\left(\frac{\pi^{2}\xi}{\mathcal{S}}\right)^{p}\frac{\Lambda_{2p}}{(2p)!}\,\frac{{\tt
Re}\;{\rm K}_{2p+2}^{{\alpha,\beta}}(i\lambda\xi)}{2p+2}$ (8)

Here $\mathcal{S}=\mathcal{M}\times\mathcal{N}$,
$\xi=\mathcal{M}/\mathcal{N}$, $\eta=(\theta_{2}\theta_{3}\theta_{4}/2)^{1/3}$
is the Dedekind-$\eta$ function; $\theta_{{\alpha,\beta}}$ are elliptic
$\theta$-functions with next relations to standard notations are
$\theta_{\frac{1}{2},\frac{1}{2}}=\theta_{3}$,
$\theta_{\frac{1}{2},0}=\theta_{4}$ and $\theta_{0,\frac{1}{2}}=\theta_{2}$,
${\rm K}_{2p+2}^{{\alpha,\beta}}(i\lambda\xi)$ are Kronecker’s double series
Izmailian2002 ; Weil , for isotropic case ($z_{v}=z_{h}=z=1$)
$\int_{0}^{\pi}\\!\\!\omega_{z}(x)~{}\\!{\rm d}x=2G$ and $G=0.915966\dots$ is
Catalan’s constant.

The differential operators $\Lambda_{2p}$ that have appeared here can be
expressed via coefficients $\lambda_{2p}$ of the Taylor expansion of the
lattice dispersion relation $\omega_{0}(k)$:

$\omega_{0}(k)=k\left(\lambda+\sum_{p=1}^{\infty}\frac{\lambda_{2p}}{(2p)!}\;k^{2p}\right)$
(9)

where $\lambda=1$, $\lambda_{2}=-2/3$, $\lambda_{4}=4$, etc. The relations
between differential operators $\Lambda_{2p}$ and coefficients $\lambda_{2p}$
can be found for example in Izmailian2002 ; Izmailian2019 .

Using the results from Izmailian2003  and by plugging the expression from
Equation (8) into Equations (3), (4), and (5), we can obtain $f_{p}(\rho)$ for
each of the three types of cylinders.

For dimers on $2M\times 2N$ cylinder one has:

$\displaystyle f_{0}(\rho)$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\ln{\frac{2\theta_{3}^{2}(\rho)}{\theta_{2}(\rho)\theta_{4}(\rho)}}=-\ln{\frac{\theta_{3}(\rho)}{\eta(\rho)}},$
(10) $\displaystyle f_{p}(\rho)$ $\displaystyle=$
$\displaystyle\frac{2^{-2p-1}{\pi}^{2p+1}\rho^{p+1}}{(2p)!(p+1)}\Lambda_{2p}\,{\tt
Re}\;{\rm K}_{p+1}^{\frac{1}{2},\frac{1}{2}}(2i\lambda\rho),$ (11)

with

$S=2M\times N\qquad\mbox{and}\qquad\rho=\frac{2M}{N}.$ (12)

For dimers on $(2M-1)\times 2N$ cylinder it has been obtained:

$\displaystyle f_{0}(\rho)$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\ln{\frac{2\theta_{4}^{2}(\rho)}{\theta_{2}(\rho)\theta_{3}(\rho)}}=-\ln{\frac{\theta_{4}(\rho)}{\eta(\rho)}},$
(13) $\displaystyle f_{p}(\rho)$ $\displaystyle=$
$\displaystyle\frac{2^{-2p-1}{\pi}^{2p+1}\rho^{p+1}}{(2p)!(p+1)}\Lambda_{2p}\,{\tt
Re}\;{\rm K}_{p+1}^{\frac{1}{2},0}(2i\lambda\rho),$ (14)

with

$S=2M\times N\qquad\mbox{and}\qquad\rho=\frac{2M}{N}.$ (15)

For dimers on $2M\times(2N-1)$ cylinder it has been received:

$\displaystyle f_{0}(\rho)$ $\displaystyle=$
$\displaystyle-\frac{1}{6}\ln{\frac{2\theta_{2}^{2}(\rho)}{\theta_{3}(\rho)\theta_{4}(\rho)}}=-\frac{1}{2}\ln{\frac{\theta_{2}(\rho)}{\eta(\rho)}},$
(16) $\displaystyle f_{p}(\rho)$ $\displaystyle=$
$\displaystyle\frac{{\pi}^{2p+1}\rho^{p+1}}{(2p)!(2p+2)}\Lambda_{2p}\,{\tt
Re}\;{\rm K}_{2p+2}^{0,\frac{1}{2}}(2i\lambda\rho),$ (17)

with

$S=2M\times 2N\qquad\mbox{and}\qquad\rho=\frac{M}{N}.$ (18)

The expressions for $\Lambda_{2p}$ and the expressions for
$K_{2p+2}^{\frac{1}{2},\frac{1}{2}}(\rho)$ and
$K_{2p+2}^{\frac{1}{2},0}(\rho)$ are given in Izmailian2019 ; Izmailian2023 .
The expressions for $K_{2p+2}^{0,\frac{1}{2}}(\rho)$ we can obtain from
relations between Kronecker’s double series $K_{2p}^{1/2,1/2}(\tau)$,
$K_{2p}^{1/2,0}(\tau)$ and $K_{2p}^{0,1/2}(\tau)$ under the transformations
$\theta_{2}\leftrightarrow\theta_{4}$ , $\theta_{3}\leftrightarrow\theta_{4}$
and $\theta_{2}\leftrightarrow\theta_{3}$, received in Appendix B and C from
Izmailian2023 . The results for $K_{2p+2}^{0,\frac{1}{2}}(\rho)$ are listed in
Supplementary Materials. Using these expressions for $\Lambda_{2p}$ and
Kronecker’s double series we can express the subleading correction terms
$f_{p}(\rho)$ in the asymptotic expansion of the free energy for the dimer
model on the cylinder in all three cases $2M\times 2N$, $(2M-1)\times 2N$ and
$2M\times(2N-1)$ for any value of $p$ in terms of the elliptic theta functions
${\theta}_{2},{\theta}_{3},{\theta}_{4}$ and the elliptic integral of the
second kind $E$. In particular in this paper we have calculated the subleading
correction terms $f_{p}$ in terms of the elliptic functions the elliptic
integral of the second kind up to $p=17$. Due to very large expressions for
$f_{p}(\rho)$ for $p>3$ we have not listed those expressions in Appendix A, B
and C for dimers on $2M\times 2N$, $(2M-1)\times 2N$ and $2M\times(2N-1)$
cylinders, respectively.

In Fig. 1, 2 and 3 we plot the behavior of the subleading correction terms
$f_{p}(\rho)$ (a) $p=0$, (b) $p=2$, (c) $p=7$, and (d) $p=14$ as a function of
the aspect ratio $\rho$ for dimers on $2M\times 2N$, $(2M-1)\times 2N$ and
$2M\times(2N-1)$ cylinders, respectively. For dimers on $2M\times 2N$ and
$(2M-1)\times 2N$ cylinders exact numerical values of $f_{p}(\rho)$, with the
aspect ratio $\rho=1,2,4$ for $p=1,2,3,...,17$ are given in Table 1 and 2,
respectively. For dimers on $2M\times(2N-1)$ cylinder exact numerical values
of $f_{p}(\rho)$, with the aspect ratio $\rho=1/2,1,2$ for $p=1,2,3,...,17$
are given in Table 3.

In Supplementary Materials we have presented the expressions of subleading
correction terms $f_{p}(\rho)$ in terms of gamma function for the aspect ratio
$\rho=1/2,1,2,4$ and $p=1,2,3,...,8$ obtained using the relation between
elliptic theta functions, elliptic integral and gamma function (see Appendix
D). The expressions for $p>8$ are so large that they are therefore not given.

We noticed that the roots coefficients $f_{p}(\rho)$ as a function of the
aspect ratio $\rho$ for increasing $p$ exponentially tends to $2$ for dimers
on $(2M-1)\times 2N$ cylinder and to $1$ for dimers on $2M\times(2N-1)$
cylinder. These results are shown in the Fig. 4. We also found that the
maximum of the function $f_{p}(\rho)$ for dimers on $2M\times 2N$ cylinder is
reached at $\rho=2$, which is also seen in Fig. 1.

Figure 1: The behavior of the correction terms $f_{p}(\rho)$ of dimers on $2M\times 2N$ cylinder for (a) $p=0$, (b) $p=2$, (c) $p=7$, and (d) $p=14$. Table 1: Coefficients $f_{p}$ in the asymptotic expansion of the free energy for model of dimers on $2M\times 2N$ cylinder, for the aspect ratio 1, 2, 4. $p$ | $f_{p}(1)$ | $f_{p}(2)$ | $f_{p}(4)$
---|---|---|---
0 | -0.346573590280… | -0.527330191080… | -1.04720452587…
1 | -0.320716568887… | -0.188074112577… | -0.320716568887…
2 | -0.810301005787… | -0.168437484866… | -0.810301005787…
3 | -6.27315437483… | -0.822513387616… | -6.27315437483…
4 | $-9.80532584302...\times 10^{1}$ | $-0.601821634381...\times 10^{1}$ | $-9.80532584302...\times 10^{1}$
5 | $-0.254149909928...\times 10^{4}$ | $-0.798824718897...\times 10^{2}$ | $-0.254149909928...\times 10^{4}$
6 | $-9.85769596813...\times 10^{4}$ | $-0.153686749144...\times 10^{4}$ | $-9.85769596813...\times 10^{4}$
7 | $-5.34679715780...\times 10^{6}$ | $-4.18016773945...\times 10^{4}$ | $-5.34679715780...\times 10^{6}$
8 | $-3.86387617973...\times 10^{8}$ | $-1.50895329542...\times 10^{6}$ | $-3.86387617973...\times 10^{8}$
9 | $-3.58814194064...\times 10^{10}$ | $-7.00865228737...\times 10^{7}$ | $-3.58814194064...\times 10^{10}$
10 | $-4.16355939979...\times 10^{12}$ | $-4.06586520894...\times 10^{9}$ | $-4.16355939979...\times 10^{12}$
11 | $-5.90325407711...\times 10^{14}$ | $-2.88247381999...\times 10^{11}$ | $-5.90325407711...\times 10^{14}$
12 | $-1.00417540827...\times 10^{17}$ | $-2.45159289451...\times 10^{13}$ | $-1.00417540827...\times 10^{17}$
13 | $-2.01840674021...\times 10^{19}$ | $-2.46387777220...\times 10^{15}$ | $-2.01840674021...\times 10^{19}$
14 | $-4.73259659515...\times 10^{21}$ | $-2.88854681755...\times 10^{17}$ | $-4.73259659515...\times 10^{21}$
15 | $-1.28024730925...\times 10^{24}$ | $-3.90700512651...\times 10^{19}$ | $-1.28024730925...\times 10^{24}$
16 | $-3.95771049508...\times 10^{26}$ | $-6.03898676038...\times 10^{21}$ | $-3.95771049508...\times 10^{26}$
17 | $-1.38650652641...\times 10^{29}$ | $-1.05782054241...\times 10^{24}$ | $-1.38650652641...\times 10^{29}$
Figure 2: The behavior of the correction terms $f_{p}(\rho)$ of dimers on $(2M-1)\times 2N$ cylinder for (a) $p=0$, (b) $p=2$, (c) $p=7$, and (d) $p=14$. Table 2: Coefficients $f_{p}$ in the asymptotic expansion of the free energy for model of dimers on $(2M-1)\times 2N$ cylinder, for the aspect ratio 1, 2, and 4. $p$ | $f_{p}(1)$ | $f_{p}(2)$ | $f_{p}(4)$
---|---|---|---
0 | -0.173286795140… | -0.519860385420… | -1.04719057650…
1 | 0.344225832959… | 0.0470185281444… | -0.282111168866…
2 | 0.805037334385… | -0.0210546856082… | -0.757968681896…
3 | 6.27827802614… | 0.0409892105392… | -6.25219341661…
4 | $9.80469113511...\times 10^{1}$ | $-0.101553266149...$ | $-9.79163137594...\times 10^{1}$
5 | $0.254151503399...\times 10^{4}$ | $0.509910660391...$ | $-0.253992195934...\times 10^{4}$
6 | $0.985769098723...\times 10^{5}$ | $-3.18777342933...$ | $-0.985635369519...\times 10^{5}$
7 | $5.34679739918...\times 10^{6}$ | $0.308966466082...\times 10^{2}$ | $-5.34665993573...\times 10^{6}$
8 | $3.86387616558...\times 10^{8}$ | $-0.362149183062...\times 10^{3}$ | $-3.86384753819...\times 10^{8}$
9 | $3.58814194176...\times 10^{10}$ | $0.577022853154...\times 10^{4}$ | $-3.58813453543...\times 10^{10}$
10 | $4.16355939969...\times 10^{12}$ | $-1.08826842309...\times 10^{5}$ | $-4.16355741264...\times 10^{12}$
11 | $5.90325407712...\times 10^{14}$ | $2.58698242660...\times 10^{6}$ | $-5.90325340194...\times 10^{14}$
12 | $1.00417540827...\times 10^{17}$ | $-7.14982909181...\times 10^{7}$ | $-1.00417537816...\times 10^{17}$
13 | $2.01840674021...\times 10^{19}$ | $2.37560308928...\times 10^{9}$ | $-2.01840672489...\times 10^{19}$
14 | $4.73259659515...\times 10^{21}$ | $-9.03636358888...\times 10^{10}$ | $-4.73259658638...\times 10^{21}$
15 | $1.28024730925...\times 10^{24}$ | $4.00370504841...\times 10^{12}$ | $-1.28024730866...\times 10^{24}$
16 | $3.95771049508...\times 10^{26}$ | $-2.00398753867...\times 10^{14}$ | $-3.95771049461...\times 10^{26}$
17 | $1.38650652641...\times 10^{29}$ | $1.14262763047...\times 10^{16}$ | $-1.38650652637...\times 10^{29}$
Figure 3: The behavior of the correction terms $f_{p}(\rho)$ of dimers on $2M\times(2N-1)$ cylinder for (a) $p=0$, (b) $p=2$, (c) $p=7$, and (d) $p=14$. Table 3: Coefficients $f_{p}$ in the asymptotic expansion of the free energy for model of dimers on $2M\times(2N-1)$ cylinder, for the aspect ratio $\frac{1}{2}$, 1, and 2. $p$ | $f_{p}(\frac{1}{2})$ | $f_{p}(1)$ | $f_{p}(2)$
---|---|---|---
0 | -0.259930192710… | -0.086643397570… | 0.177021697970…
1 | -0.282111168866… | 0.0470185281444… | 0.344225832959…
2 | -1.51593736379… | -0.0421093712164… | 1.61007466877…
3 | $-2.50087736664...\times 10^{1}$ | 0.163956842157… | $0.251131121046...\times 10^{2}$
4 | $-0.783330510075...\times 10^{3}$ | $-0.812426129192...$ | $0.784375290809...\times 10^{3}$
5 | $-0.406387513494...\times 10^{5}$ | $0.815857056626...\times 10^{1}$ | $0.406642405438...\times 10^{5}$
6 | $-0.315403318246...\times 10^{7}$ | $-0.102008749739...\times 10^{3}$ | $3.15446111591...\times 10^{6}$
7 | $-0.342186235887...\times 10^{9}$ | $0.197738538293...\times 10^{4}$ | $3.42195033547...\times 10^{8}$
8 | $-0.494572484888...\times 10^{11}$ | $-0.463550954319...\times 10^{5}$ | $4.94576149194...\times 10^{10}$
9 | $-9.18562441070...\times 10^{12}$ | $1.47717850408...\times 10^{6}$ | $9.18564337091...\times 10^{12}$
10 | $-2.13174139527...\times 10^{15}$ | $-5.57193432624...\times 10^{7}$ | $2.13174241264...\times 10^{15}$
11 | $-6.04493148359...\times 10^{17}$ | $2.64907000484...\times 10^{9}$ | $6.04493217497...\times 10^{17}$
12 | $-2.05655117448...\times 10^{20}$ | $-1.46428499800...\times 10^{11}$ | $2.05655123614...\times 10^{20}$
13 | $-8.26739394514...\times 10^{22}$ | $9.73047025371...\times 10^{12}$ | $8.26739400791...\times 10^{22}$
14 | $-3.87694312357...\times 10^{25}$ | $-7.40258905201...\times 10^{14}$ | $3.87694313074...\times 10^{25}$
15 | $-2.09755719051...\times 10^{28}$ | $6.55967035132...\times 10^{16}$ | $-2.09755719148...\times 10^{28}$
16 | $-1.29686257488...\times 10^{31}$ | $-6.56666636671...\times 10^{18}$ | $1.29686257503...\times 10^{31}$
17 | $-9.08660917119...\times 10^{33}$ | $7.48832443907...\times 10^{20}$ | $9.08660917146...\times 10^{33}$
Figure 4: The behavior of roots of the correction terms $f_{p}(\rho)$, as a
function of $p$. The dots represent our exact results. For dimers on
$(2M-1)\times 2N$ cylinder (a) the solid line is given by
$(-1)^{p}a\,b^{p}+2$, with $a=-1.08577$ and $b=0.25831$. For dimers on
$2M\times(2N-1)$ cylinder (b) the solid line is given by
$(-1)^{p+1}a\,b^{p}+1$, with $a=-0.41405$ and $b=0.29709$.

##  IV Dimer model on infinitely long strip and cylinder

Using Kronecker’s functions asymptotic form (see for example Izmailian2019 ;
Izmailian2023 ) when $\xi=\mathcal{M}/\mathcal{N}\to\infty$ (i.e.
${\mathcal{M}}\to\infty$) for fixed ${\mathcal{N}}$ from Eq. (6) one obtains
the free energy expansion for infinitely long cylinder of circumference
$2{\mathcal{N}}$ with periodic boundary conditions

$\displaystyle\lim_{{\mathcal{M}}\to\infty}\frac{F}{{\mathcal{M}}}$
$\displaystyle=$
$\displaystyle{\mathcal{N}}f_{bulk}-\frac{\pi}{24{\mathcal{N}}}+\sum_{p=1}^{\infty}\frac{f_{p}^{\mathrm{cyl}}}{{\mathcal{N}}^{2p+1}},$
(19)

where $f_{p}^{\mathrm{cyl}}$ is given by

$\displaystyle f_{p}^{\mathrm{cyl}}=\lim_{\xi\to\infty}\xi^{-p-1}f_{p}(\xi).$
(20)

In the limit $\xi\to 0$ (i.e. ${\mathcal{N}}\to\infty$) for fixed
${\mathcal{M}}$ we obtain the expansion of free energy of infinitely long
strip with free boundary condition of the width ${\mathcal{M}}$

$\displaystyle\lim_{{\mathcal{N}}\to\infty}\frac{F}{{\mathcal{N}}}$
$\displaystyle=$
$\displaystyle{\mathcal{M}}f_{bulk}+f_{1s}+\frac{\pi}{24{\mathcal{M}}}+\sum_{p=1}^{\infty}\frac{f_{p}^{\mathrm{str}}}{{\mathcal{M}}^{2p+1}},$
(21)

where $f_{p}^{\mathrm{str}}$ is given by

$\displaystyle f_{p}^{\mathrm{str}}=\lim_{\xi\to 0}\xi^{p+1}f_{p}(\xi).$ (22)

By performing the corresponding limit transitions as in Izmailian2019 ;
Izmailian2023 ; Izmailian2002a ) for the coefficients, we obtain:

for dimers on $2M\times 2N$ cylinder with $\rho=2M/N$:

$\displaystyle
f_{p}^{\mathrm{cyl}}=\frac{2^{-2p-1}\pi^{2p+1}\lambda_{2p}B_{2p+2}^{1/2}}{(2p)!(p+1)},$
(23)

$\displaystyle
f_{p}^{\mathrm{str}}=\frac{2\pi^{2p+1}\lambda_{2p}B_{2p+2}^{1/2}}{(2p)!(p+1)},$
(24)

Table 4: Coefficients $f_{p}$ for dimers on $2M\times 2N$ cylinder in the asymptotic expansion of the free energy for the infinite strip ($f_{p}^{\mathrm{str}}$) and for the infinite cylinder ($f_{p}^{\mathrm{cyl}}$). $p$ | $f_{p}^{\mathrm{str}}$ | $f_{p}^{\mathrm{cyl}}$
---|---|---
0 | -0.523598775598… | -0.130899693900…
1 | -0.301449912170… | -0.0188406195106…
2 | -0.784276639248… | -0.0122543224882…
3 | -6.26293856529… | -0.0244646037707…
4 | $-9.79850293217...\times 10^{1}$ | -0.0956885051969
5 | $-0.254071064704...\times 10^{4}$ | -0.620290685311
6 | $-0.985702483672...\times 10^{5}$ | $-0.601625051070...\times 10^{1}$
7 | $-5.34672854693...\times 10^{6}$ | $-0.815846030721...\times 10^{2}$
8 | $-3.86386185898...\times 10^{8}$ | $-0.147394632682...\times 10^{4}$
9 | $-3.58813823804...\times 10^{10}$ | $-0.342191528133...\times 10^{5}$
10 | $-4.16355840622...\times 10^{12}$ | $-0.992669679216...\times 10^{6}$
11 | $-5.90325373952...\times 10^{14}$ | $-3.51861342163...\times 10^{7}$
12 | $-1.00417539322...\times 10^{17}$ | $-1.49633794012\times 10^{9}$
13 | $-2.01840673255...\times 10^{19}$ | $-7.51915101912...\times 10^{10}$
14 | $-4.73259659077...\times 10^{21}$ | $-4.40757404153...\times 10^{12}$
15 | $-1.28024730896...\times 10^{24}$ | $-2.98080804981...\times 10^{14}$
16 | $-3.95771049485...\times 10^{26}$ | $-2.30369070478...\times 10^{16}$
17 | $-1.38650652639...\times 10^{29}$ | $-2.01763254355...\times 10^{18}$

for dimers on $(2M-1)\times 2N$ cylinder with $\rho=2M/N$:

$\displaystyle
f_{p}^{\mathrm{cyl}}=\frac{2^{-2p-1}\pi^{2p+1}\lambda_{2p}B_{2p+2}^{1/2}}{(2p)!(p+1)},$
(25)

$\displaystyle
f_{p}^{\mathrm{str}}=\frac{2\pi^{2p+1}\lambda_{2p}B_{2p+2}}{(2p)!(p+1)},$ (26)

Table 5: Coefficients $f_{p}$ for dimers on $(2M-1)\times 2N$ cylinder in the asymptotic expansion of the free energy for the infinite strip ($f_{p}^{\mathrm{str}}$) and for the infinite cylinder ($f_{p}^{\mathrm{cyl}}$). $p$ | $f_{p}^{\mathrm{str}}$ | $f_{p}^{\mathrm{cyl}}$
---|---|---
0 | 1.04719755120… | -0.130899693900…
1 | 0.344514185337… | -0.0188406195106…
2 | 0.809575885675… | -0.0122543224882…
3 | 6.31225304218… | -0.0244646037707…
4 | $9.81767808468...\times 10^{1}$ | -0.0956885051969
5 | $0.254195183445...\times 10^{4}$ | -0.620290685311
6 | $0.985822823373...\times 10^{5}$ | $-0.601625051070...\times 10^{1}$
7 | $5.34689172112...\times 10^{6}$ | $-0.815846030721...\times 10^{2}$
8 | $3.86389133814...\times 10^{8}$ | $-0.147394632682...\times 10^{4}$
9 | $3.58814508188...\times 10^{10}$ | $-0.342191528133...\times 10^{5}$
10 | $4.16356039156...\times 10^{12}$ | $-0.992669679216...\times 10^{6}$
11 | $5.90325444325...\times 10^{14}$ | $-3.51861342163...\times 10^{7}$
12 | $1.00417542314...\times 10^{17}$ | $-1.49633794012\times 10^{9}$
13 | $2.01840674759...\times 10^{19}$ | $-7.51915101912...\times 10^{10}$
14 | $4.73259659958...\times 10^{21}$ | $-4.40757404153...\times 10^{12}$
15 | $1.28024730955...\times 10^{24}$ | $-2.98080804981...\times 10^{14}$
16 | $3.95771049531...\times 10^{26}$ | $-2.30369070478...\times 10^{16}$
17 | $1.38650652643...\times 10^{29}$ | $-2.01763254355...\times 10^{18}$

for dimers on $2M\times(2N-1)$ cylinder with $\rho=M/N$:

$\displaystyle
f_{p}^{\mathrm{cyl}}=\frac{\pi^{2p+1}\lambda_{2p}B_{2p+2}}{(2p)!(2p+2)},$ (27)

$\displaystyle
f_{p}^{\mathrm{str}}=\frac{\pi^{2p+1}\lambda_{2p}B_{2p+2}^{1/2}}{(2p)!2p+2)}.$
(28)

Table 6: Coefficients $f_{p}$ for dimers on $2M\times(2N-1)$ cylinder in the asymptotic expansion of the free energy for the infinite strip ($f_{p}^{\mathrm{str}}$) and for the infinite cylinder ($f_{p}^{\mathrm{cyl}}$). $p$ | $f_{p}^{\mathrm{str}}$ | $f_{p}^{\mathrm{cyl}}$
---|---|---
0 | -0.130899693900… | 0.261799387799…
1 | -0.0753624780424… | 0.0861285463342…
2 | -0.196069159812… | 0.202393971419…
3 | -1.56573464132… | 1.57806326054…
4 | $-2.44962573304...\times 10^{1}$ | $2.45441952117...\times 10^{1}$
5 | $-0.635177661759...\times 10^{3}$ | $0.635487958614...\times 10^{3}$
6 | $-0.246425620918...\times 10^{5}$ | $0.246455705843...\times 10^{5}$
7 | $-1.33668213673...\times 10^{6}$ | $1.33672293028...\times 10^{6}$
8 | $-9.65965464746...\times 10^{7}$ | $9.65972834534...\times 10^{7}$
9 | $-8.97034559509...\times 10^{9}$ | $8.97036270470...\times 10^{9}$
10 | $-1.04088960155...\times 10^{12}$ | $1.04089009789...\times 10^{12}$
11 | $-1.47581343488...\times 10^{14}$ | $1.47581361081...\times 10^{14}$
12 | $-2.51043848305...\times 10^{16}$ | $2.51043855786...\times 10^{16}$
13 | $-5.04601683138...\times 10^{18}$ | $5.04601686897...\times 10^{18}$
14 | $-1.18314914769...\times 10^{21}$ | $1.18314914990...\times 10^{21}$
15 | $-3.20061827240...\times 10^{23}$ | $3.20061827389...\times 10^{23}$
16 | $-9.89427623712...\times 10^{25}$ | $9.89427623827...\times 10^{25}$
17 | $-3.46626631597...\times 10^{28}$ | $3.46626631607...\times 10^{28}$

We have also used in all three cases the relation of the Bernoulli numbers
$B_{n}\equiv B_{n}^{0}$ and $B_{n}^{1/2}=(2^{1-n}-1)B_{n}$.

Exact numerical values of $f_{p}^{\mathrm{cyl}}$ and $f_{p}^{\mathrm{str}}$
for dimers on $2M\times 2N$, $(2M-1)\times 2N$ and $(2M-1)\times 2N$ cylinders
are given in Table 4, 6 and 6, respectively. The exact expressions are given
for all three cases in Supplementary Materials.

##  V The ratio of the coefficients $f_{p}(\rho)$ in the free energy expansion

Now we consider ratios $r_{p}$ for the dimer model on cylinder analogous the
Izmailian2019 ; Izmailian2023 .

The ratio of the coefficients in the free energy expansion $f_{p}(\rho)$ for
$\rho<\rho_{0}$ multiplied by $\rho^{p+1}$ and the coefficients $f_{p}$ for
the strip $f_{p}^{\mathrm{str}}$ is written as

$r_{p}^{\mathrm{str}}(\rho)=\frac{\rho^{p+1}f_{p}(\rho)}{f_{p}^{\mathrm{str}}}$

and the ratio of the coefficients in the free energy expansion $f_{p}(\rho)$
for $\rho>\rho_{0}$ multiplied by $\rho^{-p-1}$ and the coefficients $f_{p}$
for the infinitely long cylinder $f_{p}^{\mathrm{cyl}}$ as

$r_{p}^{\mathrm{cyl}}(\rho)=\frac{\rho^{-p-1}f_{p}(\rho)}{f_{p}^{\mathrm{cyl}}}.$

For the dimers on $2M\times 2N$ cylinder the $\rho_{0}=2$. We list the values
of $r_{p}^{\mathrm{str}}(\rho)$ and $r_{p}^{\mathrm{cyl}}(\rho)$ for $p=0$ up
to $p=17$ in Table 7 and plot them against $p$ in Fig. 5. The behavior of
ratios depending on $\rho$ shown in Fig. 6. One can clearly see that the
ratios $r_{p}^{\mathrm{str}}(\rho)$ and $r_{p}^{\mathrm{cyl}}(\rho)$
exponentially tend to 1 in both cases as $p$ increase and $\rho\neq 2$. From
Table 7 and Figs. 5 and 6, one also sees that the ratio $r_{p}$ exponentially
tends to $2$ for $\rho=2$ and large values of $p$.

Figure 5: The behavior of ratio of the correction terms $f_{p}(\rho)$, as a function of $p$ for dimers on $2M\times 2N$ cylinder. The dots represent our exact results. For $\rho=1$ (a) the solid line is given by $a\,b^{p}+1$, with $a=0.17017$ and $b=0.38525$. For $\rho=2$ (b) and (c) the solid line is given by $(-1)^{p}a\,b^{p}+2$, with $a=-1.05881$ and $b=0.47912$. For $\rho=4$ (d) the solid line is given by $a\,b^{p}+1$., with $a=0.17017$ and $b=0.38525$.  Figure 6: The behavior of ratios of the coefficients $f_{p}(\rho)$ for dimers on $2M\times 2N$ cylinder in the asymptotic expansion of the free energy with aspect ratio $\rho<2$ times $\rho^{p+1}$ to the asymptotic coefficients for the strip $f_{p}^{\mathrm{str}}$ and the coefficients $f_{p}(\rho)$ with aspect ratio $\rho>2$ times $\rho^{-p-1}$ to the asymptotic coefficients for the cylinder $f_{p}^{\mathrm{cyl}}$, as a function of $\rho$ for p =1, 2, 3,…,17. Table 7: Ratios of the coefficients in the asymptotic expansion of the free energy $f_{p}(\rho)$ for dimers on $2M\times 2N$ cylinder with $\rho=1$ to the asymptotic coefficients for the strip $f_{p}^{\mathrm{str}}$, $f_{p}(\rho)$ with $\rho=2$ times $2^{p+1}$ to the asymptotic coefficients for the strip $f_{p}^{\mathrm{str}}$ , $f_{p}(\rho)$ with $\rho=2$ times $2^{-p-1}$ to the asymptotic coefficients for the infinite cylinder $f_{p}^{\mathrm{cyl}}$ and $f_{p}(\rho)$ with $\rho=4$ times $4^{-p-1}$ to the asymptotic coefficients for the infinite cylinder $f_{p}^{\mathrm{cyl}}$ as a function of $p$. $p$ | $r_{p}^{\mathrm{str}}(1)=\frac{f_{p}(1)}{f_{p}^{\mathrm{str}}}$ | $r_{p}^{\mathrm{str}}(2)=\frac{2^{p+1}f_{p}(2)}{f_{p}^{\mathrm{str}}}$ | $r_{p}^{\mathrm{cyl}}(2)=\frac{2^{-p-1}f_{p}(2)}{f_{p}^{\mathrm{cyl}}}$ | $r_{p}^{\mathrm{cyl}}(4)=\frac{4^{-p-1}f_{p}(4)}{f_{p}^{\mathrm{cyl}}}$
---|---|---|---|---
0 | 0.661906800458… | 2.01425295725… | 2.01425295725… | 2.00001332064…
1 | 1.06391329352… | 2.49559352960… | 2.49559352960… | 1.06391329352…
2 | 1.03318263638… | 1.71814358798… | 1.71814358798… | 1.03318263638…
3 | 1.00163115276… | 2.10128425573… | 2.10128425573… | 1.00163115276…
4 | 1.00069632177… | 1.96543211075… | 1.96543211075… | 1.00069632177…
5 | 1.00031032744… | 2.01222370871… | 2.01222370871… | 1.00031032744…
6 | 1.00006808661… | 1.99572428967… | 1.99572428967… | 1.00006808661…
7 | 1.00001283231… | 2.00145365882… | 2.00145365882… | 1.00001283231…
8 | 1.00000370633… | 1.99951270375… | 1.99951270375… | 1.00000370633…
9 | 1.00000103190… | 2.00016260973… | 2.00016260973… | 1.00000103190…
10 | 1.00000023864… | 1.99994599222… | 1.99994599222… | 1.00000023864…
11 | 1.00000005719… | 2.00001783553… | 2.00001783553… | 1.00000005719…
12 | 1.00000001499… | 1.99999413723… | 1.99999413723… | 1.00000001499…
13 | 1.00000000380… | 2.00000192075… | 2.00000192075… | 1.00000000380…
14 | 1.00000000093… | 1.99999937248… | 1.99999937248… | 1.00000000093…
15 | 1.00000000023… | 2.00000020449… | 2.00000020449… | 1.00000000023…
16 | 1.00000000006… | 1.99999993351… | 1.99999993351… | 1.00000000006…
17 | 1.00000000001… | 2.00000002157… | 2.00000002157… | 1.00000000001…

For the dimers on $(2M-1)\times 2N$ cylinder $\rho_{0}=2$. The values of
$r_{p}^{\mathrm{str}}(\rho)$ and $r_{p}^{\mathrm{cyl}}(\rho)$ for $p=0$ up to
$p=17$ are given in Table 8 and plotted them against $p$ in Fig. 7. The
dependence of the ratios on $\rho$ is shown in Fig. 8. The behavior of the
ratios is so different from what was in the previous case on the $2M\times 2N$
cylinder. Ratios $r_{p}^{\mathrm{str}}(\rho)$ and $r_{p}^{\mathrm{cyl}}(\rho)$
exponentially tend to $1$ in both cases as $p$ increase and $\rho\neq 2$. From
Table 8 and Figs. 7 and 8, one sees also that the ratio $r_{p}$ exponentially
tends to $0$ for $\rho=2$ and large values of $p$.

Figure 7: The behavior of ratio of the correction terms $f_{p}(\rho)$, as a function of $p$ for dimers on $(2M-1)\times 2N$ cylinder. The dots represent our exact results. For $\rho=1$ (a) the solid line is given by $a\,b^{p}+1$, with $a=-0.00483$ and $b=0.76631$. For $\rho=2$ the solid line is given by $(-1)^{p}a\,b^{p}$, with $a=-1.35106$ and $b=0.40254$ (b) and with $1.67625$ and $b=0.37047$ (c). For $\rho=4$ (d) the solid line is given by $a\,b^{p}+1$, with $a=-0.17005$ and $b=0.38714$.  Figure 8: The behavior of ratios of the coefficients $f_{p}(\rho)$ for dimers on $(2M-1)\times 2N$ cylinder in the asymptotic expansion of the free energy with aspect ratio $\rho<2$ times $\rho^{p+1}$ to the asymptotic coefficients for the strip $f_{p}^{\mathrm{str}}$ and the coefficients $f_{p}(\rho)$ with aspect ratio $\rho>2$ times $\rho^{-p-1}$ to the asymptotic coefficients for the cylinder $f_{p}^{\mathrm{cyl}}$, as a function of $\rho$ for p =1, 2, 3,…,17. Table 8: Ratios of the coefficients in the asymptotic expansion of the free energy $f_{p}(\rho)$ for dimers on $(2M-1)\times 2N$ cylinder with $\rho=1$ to the asymptotic coefficients for the strip $f_{p}^{\mathrm{str}}$, $f_{p}(\rho)$ with $\rho=2$ times $2^{p+1}$ to the asymptotic coefficients for the strip $f_{p}^{\mathrm{str}}$ , $f_{p}(\rho)$ with $\rho=2$ times $2^{-p-1}$ to the asymptotic coefficients for the infinite cylinder $f_{p}^{\mathrm{cyl}}$ and $f_{p}(\rho)$ with $\rho=4$ times $4^{-p-1}$ to the asymptotic coefficients for the infinite cylinder $f_{p}^{\mathrm{cyl}}$ as a function of $p$. $p$ | $r_{p}^{\mathrm{str}}(1)=\frac{f_{p}(1)}{f_{p}^{\mathrm{str}}}$ | $r_{p}^{\mathrm{str}}(2)=\frac{2^{p+1}f_{p}(2)}{f_{p}^{\mathrm{str}}}$ | $r_{p}^{\mathrm{cyl}}(2)=\frac{2^{-p-1}f_{p}(2)}{f_{p}^{\mathrm{cyl}}}$ | $r_{p}^{\mathrm{cyl}}(4)=\frac{4^{-p-1}f_{p}(4)}{f_{p}^{\mathrm{cyl}}}$
---|---|---|---|---
0 | -0.165476700114… | -0.992860200687… | 1.98572040137… | 1.99998667931…
1 | 0.999163017403… | 0.545911084601… | -0.623898382401… | 0.935847573601…
2 | 0.994393914924… | -0.208056450107… | 0.214767948497… | 0.966455768238…
3 | 0.994617608672… | 0.103897509217… | -0.104715599840… | 0.998284327945…
4 | 0.998677187269… | -0.0331005405631… | 0.0331653165720… | 0.999298713663…
5 | 0.999828163357… | 0.0128382771942… | -0.0128445489466… | 0.999689579883…
6 | 0.999945502734… | -0.00413902974531… | 0.00413953505965… | 0.999931912363…
7 | 0.999982359481… | 0.00147927841899… | -0.00147932356436… | 0.999987167629…
8 | 0.999996073245… | -0.000479879907330… | 0.000479883568551… | 0.999996293657…
9 | 0.999999124863… | 0.000164673219211… | -0.000164673533301… | 0.999998968098…
10 | 0.999999761774… | -0.0000535304768250… | 0.0000535305023503… | 0.999999761363…
11 | 0.999999937979… | 0.0000179498954708… | -0.0000179498976106… | 0.999999942814…
12 | 0.999999985190… | $-0.583278564384...\times 10^{-7}$ | $0.58327858177...\times 10^{-7}$ | 0.999999985007…
13 | 0.999999996345… | $0.192834675475...\times 10^{-7}$ | $-0.19283467691192...\times 10^{-7}$ | 0.999999996204…
14 | 0.999999999063… | $-0.625668289807...\times 10^{-8}$ | $0.625668290972...\times 10^{-8}$ | 0.999999999074…
15 | 0.99999999977… | $0.204950099949...\times 10^{-8}$ | $-0.204950100045...\times 10^{-8}$ | 0.999999999769…
16 | 0.99999999994… | $-0.663683346672...\times 10^{-9}$ | $0.663683346749...\times 10^{-9}$ | 0.999999999942…
17 | 0.99999999999… | $0.216034307703...\times 10^{-9}$ | $-0.216034307709...\times 10^{-9}$ | 0.999999999985…

For the dimers on $2M\times(2N-1)$ cylinder $\rho_{0}=1$. The results for the
ratios $r_{p}^{\mathrm{str}}(\rho)$ and $r_{p}^{\mathrm{cyl}}(\rho)$ for $p=0$
up to $p=17$ are shown in Table 9 and Fig. 9. The ratios as a function of
$\rho$ are demonstrated in Fig. 10. In this case, the behavior of the ratios
is very similar to what was in the previous case on $(2M-1)\times 2N$
cylinder, and differs only in the numerical values. Ratios
$r_{p}^{\mathrm{str}}(\rho)$ and $r_{p}^{\mathrm{cyl}}(\rho)$ exponentially
tend to $1$ in both cases as $p$ increase and $\rho\neq 1$. From Table 9 and
Figs. 9 and 10, one also sees that the ratio $r_{p}$ exponentially tends to
$0$ for $\rho=1$ and large values of $p$.

Figure 9: The behavior of ratio of the correction terms $f_{p}(\rho)$, as a function of $p$ for dimers on $2M\times(2N-1)$ cylinder. The dots represent our exact results. For $\rho=1/2$ (a) the solid line is given by $a\,b^{p}+1$, with $a=-0.17005$ and $b=0.38714$. For $\rho=1$ the solid line is given by $(-1)^{p}a\,b^{p}$, with $a=1.67625$ and $b=0.37047$ (b) and with $-1.35106$ and $b=0.40254$ (c). For $\rho=2$ (d) the solid line is given by $a\,b^{p}+1$, with $a=-0.00483$ and $b=0.76631$.  Figure 10: The behavior of ratios of the coefficients $f_{p}(\rho)$ for dimers on $2M\times(2N-1)$ cylinder in the asymptotic expansion of the free energy with aspect ratio $\rho<1$ times $\rho^{p+1}$ to the asymptotic coefficients for the strip $f_{p}^{\mathrm{str}}$ and the coefficients $f_{p}(\rho)$ with aspect ratio $\rho>1$ times $\rho^{-p-1}$ to the asymptotic coefficients for the cylinder $f_{p}^{\mathrm{cyl}}$, as a function of $\rho$ for p =1, 2, 3,…,17. Table 9: Ratios of the coefficients in the asymptotic expansion of the free energy $f_{p}(\rho)$ for dimers on $2M\times(2N-1)$ cylinder with $\rho=\frac{1}{2}$ times $(\frac{1}{2})^{p+1}$ to the asymptotic coefficients for the strip $f_{p}^{\mathrm{str}}$, $f_{p}(\rho)$ with $\rho=1$ to the asymptotic coefficients for the strip $f_{p}^{\mathrm{str}}$, $f_{p}(\rho)$ with $\rho=1$ to the asymptotic coefficients for the infinite cylinder $f_{p}^{\mathrm{cyl}}$, $f_{p}(\rho)$ and $f_{p}(\rho)$ with $\rho=2$ times $2^{-p-1}$ to the asymptotic coefficients for the infinite cylinder $f_{p}^{\mathrm{cyl}}$ as a function of $p$. $p$ | $r_{p}^{\mathrm{str}}(\frac{1}{2})=\frac{(\frac{1}{2})^{p+1}f_{p}(\frac{1}{2})}{f_{p}^{\mathrm{str}}}$ | $r_{p}^{\mathrm{str}}(1)=\frac{f_{p}(1)}{f_{p}^{\mathrm{str}}}$ | $r_{p}^{\mathrm{cyl}}(1)=\frac{f_{p}(1)}{f_{p}^{\mathrm{cyl}}}$ | $r_{p}^{\mathrm{cyl}}(2)=\frac{2^{-p-1}f_{p}(2)}{f_{p}^{\mathrm{cyl}}}$
---|---|---|---|---
0 | 0.992860200687… | 0.661906800458… | -0.330953400229… | 0.338086539197…
1 | 0.935847573601… | -0.623898382401… | 0.545911084601… | 0.999163017403…
2 | 0.966455768238… | 0.214767948497… | -0.208056450107… | 0.994393914924…
3 | 0.998284327945… | -0.104715599840… | 0.103897509217… | 0.994617608672…
4 | 0.999298713663… | 0.0331653165720… | -0.0331005405631… | 0.998677187269…
5 | 0.999689579883… | -0.0128445489466… | 0.0128382771942… | 0.999828163357…
6 | 0.999931912363… | 0.00413953505965… | -0.00413902974531… | 0.999945502734…
7 | 0.999987167629… | -0.00147932356436… | 0.00147927841899… | 0.999982359481…
8 | 0.999996293657… | 0.000479883568551… | -0.000479879907330… | 0.999996073245…
9 | 0.999998968098… | -0.000164673533301… | 0.000164673219211… | 0.999999124863…
10 | 0.999999761363… | 0.0000535305023503… | -0.000053530476825… | 0.999999761774…
11 | 0.999999942814… | -0.0000179498976106… | 0.0000179498954708… | 0.999999937979…
12 | 0.999999985007… | $0.583278581767...\times 10^{-7}$ | $-0.583278564384...\times 10^{-7}$ | 0.999999985190…
13 | 0.999999996204… | $-0.192834676912...\times 10^{-7}$ | $0.192834675475...\times 10^{-7}$ | 0.999999996345…
14 | 0.999999999074… | $0625668290972...\times 10^{-8}$ | $-0.625668289807...\times 10^{-8}$ | 0.999999999063…
15 | 0.999999999769… | $-0204950100045...\times 10^{-8}$ | $0.204950099949...\times 10^{-8}$ | 0.99999999977…
16 | 0.999999999942… | $0.663683346749...\times 10^{-9}$ | $-0.663683346672...\times 10^{-9}$ | 0.99999999994…
17 | 0.999999999985… | $-0.216034307709...\times 10^{-9}$ | $0.216034307703...\times 10^{-9}$ | 0.99999999999…

##  VI Conclusions

The partition function of the dimer model on a cylinder has been analyzed in
three cases: $2M\times 2N$, $(2M-1)\times 2N$ and $2M\times(2N-1)$, when the
lattice is completely covered by dimers. Using the results of Ref.
Izmailian2003 ; Izmailian2002  the exact expressions for the correction terms
$f_{p}(\rho)$ in the asymptotic expansion of the free energy have been
obtained from $p=0$ up to $p=17$. Asymptotic formulas for the correction terms
have been derived also as a simple exact expressions for the strip
($N\rightarrow\infty$) and for the infinitely long cylinder
($M\rightarrow\infty$).

The ratio of the coefficients $f_{p}(\rho)$ in the free energy expansion of
the dimer model cylinder in all three cases multiplied by $\rho^{p+1}$ for
$\rho<\rho_{0}$, and the coefficients $f_{p}^{\mathrm{str}}$ in the free
energy expansion for the band tends to $1$ with an increase in $p$. Likewise
the ratio of the coefficients $f_{p}(\rho)$ multiplied by $\rho^{-p-1}$ for
$\rho>\rho_{0}$ and the coefficients $f_{p}^{\mathrm{cyl}}$ in the free energy
expansion for an infinitely long cylinder tends to $1$ in the infinite $p$.
For dimers on $2M\times 2N$ and $(2M-1)\times 2N$ cylinders $\rho_{0}=2$ and
for dimer on $2M\times(2N-1)$ cylinder $\rho_{0}=1$.

Both the $r_{p}^{\text{str}}$ and $r_{p}^{\text{cyl}}$ ratios show an abrupt
change at $\rho=\rho_{0}$ for all three types of cylinders. For dimers on
$2M\times 2N$ cylinder the ratios tends to $2$, and for dimers on
$(2M-1)\times 2N$ and $2M\times(2N-1)$ cylinders, they tends to $0$. Thus, we
have similarities with the dimer case on a rectangle under free boundary
conditions, as well as with the Ising model under Braskamp-Kunz boundary
conditions in that there is an abrupt change. However, the limiting values of
the ratios and the critical values of the aspect ratio at which this occurs
differ.

As we can see in Fig. 4, the critical values of the aspect ratio $\rho_{0}$
are equal to the limiting values of the roots of the correction terms
$f_{p}(\rho)$, which for dimers on $(2M-1)\times 2N$ cylinder is equal to two,
and for dimers on $2M\times(2N-1)$ cylinder is equal to one. For dimers on the
$2M\times 2N$ cylinder, this critical value coincides with the point where
$f_{p}(\rho)$ reaches its maximum at $\rho=2$ (see Fig. 1).

The similarity in the behavior of the $r_{p}^{str}$ and $r_{p}^{cyl}$ ratios
for the various models discussed in this paper and in references Izmailian2019
; Izmailian2023  appears to be due to the fact that the finite-size correction
coefficients $f_{p}(\rho)$ are derived from Kronecker double series ${\rm
K}_{2p+2}^{\alpha,\beta}$.

The $K_{2p+2}^{\frac{1}{2},\frac{1}{2}}(\rho)$ determines finite-size
correction coefficients $f_{p}(\rho)$ for dimers on the $2M\times 2N$ cylinder
and rectangle with free boundary conditions Izmailian2019 .

The $K_{2p+2}^{\frac{1}{2},0}(\rho)$ determines finite-size correction
coefficients $f_{p}(\rho)$ for dimers on the $(2M-1)\times 2N$ cylinder and
Ising model wtih Braskamp-Kunz boundary conditions Izmailian2023 .

The $K_{2p+2}^{0,\frac{1}{2}}(\rho)$ determines finite-size correction
coefficients $f_{p}(\rho)$ for dimers on the $2M\times(2N-1)$ cylinder.

## Acknowledgement

It is with deep appreciation and gratitude that I remember my friend and
colleague, Nikolai Izmailian. He was one of the main driving forces behind
this series of papers on finite correction factors. Sadly, he is no longer
with us, and I had to complete this paper on my own.

##  Appendix A Expressions of $f_{p}(\rho)$ for dimers on $2M\times 2N$
cylinder from $p=0$ up to $p=3$.

$\displaystyle f_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\log\left(\frac{2\theta_{3}^{2}}{\theta_{2}\theta_{4}}\right)$
(29) $\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{48}\pi^{3}\rho^{2}\left(\frac{\theta_{2}^{8}}{30}-\frac{1}{2}\theta_{3}^{2}\theta_{2}^{6}+\frac{1}{30}\theta_{4}^{4}\theta_{2}^{4}-\frac{1}{4}\theta_{3}^{2}\theta_{4}^{4}\theta_{2}^{2}-\frac{7\theta_{4}^{8}}{240}\right)$
(30) $\displaystyle f_{2}$ $\displaystyle=$
$\displaystyle\frac{\pi^{5}\rho^{3}}{774144}\left(\theta_{4}^{12}\left(-42\theta_{2}^{2}\rho\text{E}+\theta_{3}^{2}\rho\left(62\text{E}+105\pi\theta_{2}^{2}\right)-83\pi\theta_{2}^{4}\rho-31\right)\right.$
(31) $\displaystyle-$
$\displaystyle\left.2\theta_{2}^{2}\theta_{4}^{8}\left(105\theta_{3}^{4}\rho\text{E}-3\theta_{3}^{2}\left(26\theta_{2}^{2}\rho\left(\text{E}+7\pi\theta_{2}^{2}\right)+21\right)+\theta_{2}^{2}\left(1029\theta_{2}^{2}\rho\text{E}+34\pi\theta_{2}^{4}\rho+39\right)\right)\right.$
(32) $\displaystyle-$
$\displaystyle\left.16\theta_{2}^{6}\theta_{4}^{4}\left(126\theta_{3}^{4}\rho\text{E}-3\theta_{3}^{2}\left(2\theta_{2}^{2}\rho\text{E}+21\pi\theta_{2}^{4}\rho+42\right)+\theta_{2}^{2}\left(336\theta_{2}^{2}\rho\text{E}+\pi\theta_{2}^{4}\rho+3\right)\right)\right.$
(33) $\displaystyle-$
$\displaystyle\left.32\theta_{2}^{10}\left(105\theta_{2}^{4}\rho\text{E}+21\theta_{3}^{4}\rho\text{E}-\theta_{3}^{2}\left(2\theta_{2}^{2}\rho\text{E}+63\right)+\theta_{2}^{2}\right)-31\pi\theta_{4}^{16}\rho\right)$
(34) $\displaystyle f_{3}$ $\displaystyle=$
$\displaystyle\frac{\pi^{7}\rho^{4}}{3715891200\theta_{3}^{2}}\left(-70\pi^{2}\rho^{2}\left(70\theta_{2}^{2}+381\theta_{3}^{2}\right)\theta_{4}^{24}-70\rho\left(7035\pi^{2}\rho\theta_{2}^{6}+1093\pi^{2}\rho\theta_{3}^{2}\theta_{2}^{4}\right.\right.$
(35) $\displaystyle+$
$\displaystyle\left.\left.5\left(6\rho\text{E}^{2}-60\pi\rho\theta_{3}^{2}\text{E}+\pi\left(41\pi\rho\theta_{3}^{4}+18\right)\right)\theta_{2}^{2}+254\theta_{3}^{2}\left(\rho\text{E}\left(\text{E}-6\pi\theta_{3}^{2}\right)+3\pi\right)\right)\theta_{4}^{20}\right.$
(36) $\displaystyle-$
$\displaystyle\left.\left(30099\theta_{3}^{2}+70\rho\left(5\left(846\rho\text{E}^{2}+\pi\left(5083\pi\rho\theta_{2}^{4}+2502\right)\right)\theta_{2}^{6}+2\left(407\rho\text{E}^{2}\theta_{2}^{2}\right.\right.\right.\right.$
(37) $\displaystyle+$
$\displaystyle\left.\left.\left.\left.\pi\left(556\pi\rho\theta_{2}^{4}+885\right)\theta_{2}^{2}-30\text{E}\left(605\pi\rho\theta_{2}^{4}+6\right)\right)\theta_{3}^{2}\theta_{2}^{2}+10\rho\text{E}\left(127\text{E}-96\pi\theta_{2}^{2}\right)\theta_{3}^{6}\right.\right.\right.$
(38) $\displaystyle+$
$\displaystyle\left.\left.\left.\left(510\rho\text{E}^{2}\theta_{2}^{2}+5\pi\left(2423\pi\rho\theta_{2}^{4}+108\right)\theta_{2}^{2}-12\text{E}\left(295\pi\rho\theta_{2}^{4}+127\right)\right)\theta_{3}^{4}\right)\right)\theta_{4}^{16}\right.$
(39) $\displaystyle-$
$\displaystyle\left.4\theta_{2}^{2}\left(175\rho\left(2472\rho\text{E}^{2}+\pi\left(3137\pi\rho\theta_{2}^{4}+4914\right)\right)\theta_{2}^{8}+28\left(5\rho\left(122\rho\text{E}^{2}\theta_{2}^{2}\right.\right.\right.\right.$
(40) $\displaystyle+$
$\displaystyle\left.\left.\left.\left.2\pi\left(29\pi\rho\theta_{2}^{4}+81\right)\theta_{2}^{2}-45\text{E}\left(380\pi\rho\theta_{2}^{4}+277\right)\right)\theta_{2}^{2}+474\right)\theta_{3}^{2}\theta_{2}^{2}+15750\rho^{2}\text{E}^{2}\theta_{3}^{8}\right.\right.$
(41) $\displaystyle+$
$\displaystyle\left.\left.140\rho\text{E}\left(266\rho\text{E}\theta_{2}^{2}-45\left(248\pi\rho\theta_{2}^{4}+3\right)\right)\theta_{3}^{6}+15\left(7\rho\left(4165\pi^{2}\rho\theta_{2}^{6}-432\pi\rho\text{E}\theta_{2}^{4}\right.\right.\right.\right.$
(42) $\displaystyle+$
$\displaystyle\left.\left.\left.\left.90\left(167\rho\text{E}^{2}+93\pi\right)\theta_{2}^{2}-448\text{E}\right)\theta_{2}^{2}+474\right)\theta_{3}^{4}\right)\theta_{4}^{12}-16\theta_{2}^{6}\left(525\rho\left(443\rho\text{E}^{2}\right.\right.\right.$
(43) $\displaystyle+$
$\displaystyle\left.\left.\left.17\pi\left(7\pi\rho\theta_{2}^{4}+30\right)\right)\theta_{2}^{8}+2\left(711-70\rho\theta_{2}^{2}\left(-2\pi^{2}\rho\theta_{2}^{6}+5190\pi\rho\text{E}\theta_{2}^{4}+\left(47\rho\text{E}^{2}+3\pi\right)\theta_{2}^{2}\right.\right.\right.\right.$
(44) $\displaystyle+$
$\displaystyle\left.\left.\left.\left.16875\text{E}\right)\right)\theta_{3}^{2}\theta_{2}^{2}+458850\rho^{2}\text{E}^{2}\theta_{3}^{8}+140\rho\text{E}\left(23\rho\text{E}\theta_{2}^{2}-45\left(152\pi\rho\theta_{2}^{4}+69\right)\right)\theta_{3}^{6}\right.\right.$
(45) $\displaystyle+$
$\displaystyle\left.\left.5\left(7\rho\left(59145\rho\text{E}^{2}\theta_{2}^{2}+5\pi\left(293\pi\rho\theta_{2}^{4}+3078\right)\theta_{2}^{2}+24\text{E}\left(\pi\rho\theta_{2}^{4}-6\right)\right)\theta_{2}^{2}+49059\right)\theta_{3}^{4}\right)\theta_{4}^{8}\right.$
(46) $\displaystyle-$
$\displaystyle\left.32\theta_{2}^{10}\left(2975\rho\left(36\rho\text{E}^{2}+\pi\left(\pi\rho\theta_{2}^{4}+18\right)\right)\theta_{2}^{8}-12\left(35\rho\left(340\pi\rho\text{E}\theta_{2}^{4}+2\left(9\rho\text{E}^{2}+\pi\right)\theta_{2}^{2}\right.\right.\right.\right.$
(47) $\displaystyle+$
$\displaystyle\left.\left.\left.\left.4335\text{E}\right)\theta_{2}^{2}+158\right)\theta_{3}^{2}\theta_{2}^{2}+410550\rho^{2}\text{E}^{2}\theta_{3}^{8}-420\rho\text{E}\left(6\rho\text{E}\theta_{2}^{2}+85\left(8\pi\rho\theta_{2}^{4}+9\right)\right)\theta_{3}^{6}\right.\right.$
(48) $\displaystyle+$
$\displaystyle\left.\left.5\left(7\rho\left(47430\rho\text{E}^{2}\theta_{2}^{2}-85\pi\left(\pi\rho\theta_{2}^{4}-54\right)\theta_{2}^{2}+48\text{E}\left(\pi\rho\theta_{2}^{4}+4\right)\right)\theta_{2}^{2}+72522\right)\theta_{3}^{4}\right)\theta_{4}^{4}\right.$
(49) $\displaystyle+$
$\displaystyle\left.384\theta_{2}^{14}\left(-2975\rho^{2}\text{E}^{2}\theta_{2}^{8}+\left(280\rho\text{E}\left(\rho\text{E}\theta_{2}^{2}+255\right)\theta_{2}^{2}+79\right)\theta_{3}^{2}\theta_{2}^{2}\right.\right.$
(50) $\displaystyle-$
$\displaystyle\left.\left.5\left(7\rho\text{E}\left(1955\rho\text{E}\theta_{2}^{2}+8\right)\theta_{2}^{2}+4029\right)\theta_{3}^{4}\right)\right)$
(51)

##  Appendix B Expressions of $f_{p}(\rho)$ for dimers on $(2M-1)\times 2N$
cylinder from $p=0$ up to $p=3$.

$\displaystyle f_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}\log\left(\frac{2\theta_{4}^{2}}{\theta_{2}\theta_{3}}\right)$
(53) $\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{48}\pi^{3}\rho^{2}\left(\frac{\theta_{2}^{8}}{30}+\frac{1}{2}\theta_{3}^{2}\theta_{2}^{6}+\frac{1}{30}\theta_{4}^{4}\theta_{2}^{4}+\frac{1}{4}\theta_{3}^{2}\theta_{4}^{4}\theta_{2}^{2}-\frac{7\theta_{4}^{8}}{240}\right)$
(54) $\displaystyle f_{2}$ $\displaystyle=$
$\displaystyle-\frac{\pi^{5}\rho^{3}}{774144}\left(-3360\theta_{2}^{14}\rho\text{E}+16\theta_{2}^{12}\left(-4\theta_{3}^{2}\rho\text{E}+\pi\theta_{4}^{4}\rho+2\right)-336\theta_{2}^{10}\left(\theta_{4}^{4}\rho\left(16\text{E}-\pi\theta_{3}^{2}\right)\right.\right.$
(55) $\displaystyle-$
$\displaystyle\left.\left.2\theta_{3}^{2}\left(-\theta_{3}^{2}\rho\text{E}+\pi\theta_{4}^{4}\rho+3\right)\right)+4\theta_{4}^{4}\theta_{2}^{8}\left(-24\theta_{3}^{2}\rho\text{E}+17\pi\theta_{4}^{4}\rho+12\right)\right.$
(56) $\displaystyle+$
$\displaystyle\left.42\theta_{4}^{4}\theta_{2}^{6}\left(-32\theta_{3}^{4}\rho\text{E}+2\theta_{3}^{2}\left(-8\theta_{3}^{2}\rho\text{E}+9\pi\theta_{4}^{4}\rho+24\right)+\theta_{4}^{4}\rho\left(8\pi\theta_{3}^{2}-49\text{E}\right)\right)\right.$
(57) $\displaystyle+$
$\displaystyle\left.\theta_{4}^{8}\theta_{2}^{4}\left(-156\theta_{3}^{2}\rho\text{E}+83\pi\theta_{4}^{4}\rho+78\right)+21\theta_{4}^{8}\theta_{2}^{2}\left(-8\theta_{3}^{4}\rho\text{E}+\theta_{3}^{2}\left(-2\theta_{3}^{2}\rho\text{E}+4\pi\theta_{4}^{4}\rho+6\right)\right.\right.$
(58) $\displaystyle+$
$\displaystyle\left.\left.\theta_{4}^{4}\rho\left(\pi\theta_{3}^{2}-2\text{E}\right)\right)+31\theta_{4}^{12}\left(-2\theta_{3}^{2}\rho\text{E}+\pi\theta_{4}^{4}\rho+1\right)\right)$
(59) $\displaystyle f_{3}$ $\displaystyle=$
$\displaystyle\frac{\pi^{7}\rho^{4}}{11147673600\theta_{3}^{2}}\left(571200\pi^{2}\rho^{2}\theta_{4}^{4}\theta_{2}^{22}+8960\pi^{2}\rho^{2}\theta_{3}^{2}\theta_{4}^{4}\theta_{2}^{20}+285600\pi\rho\theta_{4}^{4}\left(\pi\rho\left(5\theta_{3}^{4}+14\theta_{4}^{4}\right)\right.\right.$
(60) $\displaystyle+$
$\displaystyle\left.\left.18\right)\theta_{2}^{18}+128\theta_{3}^{2}\left(35\pi\rho\left(-2\pi\rho\theta_{3}^{4}+3\pi\rho\theta_{4}^{4}+18\right)\theta_{4}^{4}+711\right)\theta_{2}^{16}+120\left(65625\pi^{2}\rho^{2}\theta_{4}^{12}\right.\right.$
(61) $\displaystyle+$
$\displaystyle\left.\left.70\pi\rho\left(803\pi\rho\theta_{3}^{4}+1530\right)\theta_{4}^{8}-2380\pi\rho\theta_{3}^{4}\left(7\pi\rho\theta_{3}^{4}-54\right)\theta_{4}^{4}+193392\theta_{3}^{4}\right)\theta_{2}^{14}\right.$
(62) $\displaystyle-$
$\displaystyle\left.32\theta_{3}^{2}\theta_{4}^{4}\left(35\pi\rho\theta_{4}^{4}\left(16\pi\rho\theta_{3}^{4}+77\pi\rho\theta_{4}^{4}-18\right)-5688\right)\theta_{2}^{12}+60\theta_{4}^{4}\left(100940\pi^{2}\rho^{2}\theta_{4}^{12}\right.\right.$
(63) $\displaystyle+$
$\displaystyle\left.\left.735\pi\rho\left(197\pi\rho\theta_{3}^{4}+234\right)\theta_{4}^{8}+420\pi\rho\theta_{3}^{4}\left(1026-119\pi\rho\theta_{3}^{4}\right)\theta_{4}^{4}+580176\theta_{3}^{4}\right)\theta_{2}^{10}\right.$
(64) $\displaystyle-$
$\displaystyle\left.16\theta_{3}^{2}\theta_{4}^{8}\left(35\pi\rho\left(-12\pi\rho\theta_{3}^{4}+457\pi\rho\theta_{4}^{4}+486\right)\theta_{4}^{4}+4266\right)\theta_{2}^{8}+30\theta_{4}^{8}\left(54145\pi^{2}\rho^{2}\theta_{4}^{12}\right.\right.$
(65) $\displaystyle+$
$\displaystyle\left.\left.35\pi\rho\left(3257\pi\rho\theta_{3}^{4}+2502\right)\theta_{4}^{8}+210\pi\rho\theta_{3}^{4}\left(1674-161\pi\rho\theta_{3}^{4}\right)\theta_{4}^{4}+392472\theta_{3}^{4}\right)\theta_{2}^{6}\right.$
(66) $\displaystyle-$
$\displaystyle\left.28\theta_{3}^{2}\theta_{4}^{12}\left(5\pi\rho\left(-112\pi\rho\theta_{3}^{4}+1815\pi\rho\theta_{4}^{4}+2655\right)\theta_{4}^{4}+5688\right)\theta_{2}^{4}+30\theta_{4}^{12}\left(525\pi^{2}\rho^{2}\theta_{4}^{12}\right.\right.$
(67) $\displaystyle+$
$\displaystyle\left.\left.35\pi\rho\left(47\pi\rho\theta_{3}^{4}+18\right)\theta_{4}^{8}+35\pi\rho\theta_{3}^{4}\left(108-7\pi\rho\theta_{3}^{4}\right)\theta_{4}^{4}+2844\theta_{3}^{4}\right)\theta_{2}^{2}\right.$
(68) $\displaystyle-$
$\displaystyle\left.127\theta_{3}^{2}\theta_{4}^{16}\left(70\pi\rho\left(-\pi\rho\theta_{3}^{4}+10\pi\rho\theta_{4}^{4}+18\right)\theta_{4}^{4}+711\right)+280\rho^{2}\text{E}^{2}\left(8160\theta_{2}^{22}+1152\theta_{3}^{2}\theta_{2}^{20}\right.\right.$
(69) $\displaystyle+$
$\displaystyle\left.\left.4080\left(70\theta_{3}^{4}+5\theta_{4}^{4}\right)\theta_{2}^{18}+2560\theta_{3}^{2}\theta_{4}^{4}\theta_{2}^{16}+60\theta_{4}^{4}\left(9588\theta_{3}^{4}+273\theta_{4}^{4}\right)\theta_{2}^{14}+32\left(35\theta_{3}^{2}\theta_{4}^{8}\right.\right.\right.$
(70) $\displaystyle+$
$\displaystyle\left.\left.\left.28\theta_{3}^{6}\theta_{4}^{4}\right)\theta_{2}^{12}+30\theta_{4}^{4}\left(4896\theta_{3}^{8}+11862\theta_{4}^{4}\theta_{3}^{4}+139\theta_{4}^{8}\right)\theta_{2}^{10}-16\theta_{4}^{8}\left(36\theta_{3}^{6}+39\theta_{4}^{4}\theta_{3}^{2}\right)\theta_{2}^{8}\right.\right.$
(71) $\displaystyle+$
$\displaystyle\left.\left.30\theta_{4}^{8}\left(2760\theta_{3}^{8}+2219\theta_{4}^{4}\theta_{3}^{4}+\theta_{4}^{8}\right)\theta_{2}^{6}-\theta_{4}^{12}\left(1680\theta_{3}^{6}+463\theta_{4}^{4}\theta_{3}^{2}\right)\theta_{2}^{4}+30\theta_{3}^{2}\theta_{4}^{12}\left(24\theta_{3}^{6}\right.\right.\right.$
(72) $\displaystyle+$
$\displaystyle\left.\left.\left.12\theta_{4}^{4}\theta_{3}^{2}\right)\theta_{2}^{2}-127\theta_{3}^{2}\theta_{4}^{16}\left(8\theta_{3}^{4}+\theta_{4}^{4}\right)\right)-140\rho\text{E}\left(128\pi\rho\theta_{4}^{4}\theta_{2}^{20}+4080\left(3\left(5\pi\rho\theta_{4}^{4}+42\right)\theta_{3}^{2}\right.\right.\right.$
(73) $\displaystyle+$
$\displaystyle\left.\left.\left.2\left(8\pi\rho\theta_{4}^{4}+9\right)\theta_{3}^{2}\right)\theta_{2}^{18}+128\left(3\pi\rho\theta_{4}^{8}+8\pi\rho\theta_{3}^{4}\theta_{4}^{4}+18\theta_{3}^{4}\right)\theta_{2}^{16}+120\theta_{4}^{4}\left(1394\pi\rho\theta_{3}^{6}\right.\right.\right.$
(74) $\displaystyle+$
$\displaystyle\left.\left.\left.51\left(11\pi\rho\theta_{4}^{4}+174\right)\theta_{3}^{2}+10\left(211\pi\rho\theta_{4}^{4}+153\right)\theta_{3}^{2}\right)\theta_{2}^{14}+32\theta_{4}^{4}\left(5\pi\rho\theta_{4}^{8}+\pi\rho\theta_{3}^{4}\theta_{4}^{4}\right.\right.\right.$
(75) $\displaystyle+$
$\displaystyle\left.\left.\left.144\theta_{3}^{4}\right)\theta_{2}^{12}+90\theta_{4}^{4}\left(4\left(793\pi\rho\theta_{4}^{4}+612\right)\theta_{3}^{6}+\theta_{4}^{4}\left(91\pi\rho\theta_{4}^{4}+7362\right)\theta_{3}^{2}+18\theta_{4}^{4}\left(157\pi\rho\theta_{4}^{4}\right.\right.\right.\right.$
(76) $\displaystyle+$
$\displaystyle\left.\left.\left.\left.91\right)\theta_{3}^{2}\right)\theta_{2}^{10}-16\theta_{4}^{8}\left(20\pi\rho\theta_{4}^{8}+237\pi\rho\theta_{3}^{4}\theta_{4}^{4}+108\theta_{3}^{4}\right)\theta_{2}^{8}+15\theta_{4}^{8}\left(6\left(1327\pi\rho\theta_{4}^{4}+1656\right)\theta_{3}^{6}\right.\right.\right.$
(77) $\displaystyle+$
$\displaystyle\left.\left.\left.\theta_{4}^{4}\left(139\pi\rho\theta_{4}^{4}+7470\right)\theta_{3}^{2}+18\theta_{4}^{4}\left(248\pi\rho\theta_{4}^{4}+139\right)\theta_{3}^{2}\right)\theta_{2}^{6}-\theta_{4}^{12}\left(351\pi\rho\theta_{4}^{8}+5086\pi\rho\theta_{3}^{4}\theta_{4}^{4}\right.\right.\right.$
(78) $\displaystyle+$
$\displaystyle\left.\left.\left.4032\theta_{3}^{4}\right)\theta_{2}^{4}+15\theta_{4}^{12}\left(\left(89\pi\rho\theta_{4}^{4}+108\right)\theta_{3}^{6}+\theta_{4}^{4}\left(\pi\rho\theta_{4}^{4}+18\right)\theta_{3}^{2}\right.\right.\right.$
(79) $\displaystyle+$
$\displaystyle\left.\left.\left.18\left(2\pi\rho\theta_{4}^{8}+\theta_{4}^{4}\right)\theta_{3}^{2}\right)\theta_{2}^{2}-127\theta_{4}^{16}\left(18\theta_{3}^{4}+\pi\rho\theta_{4}^{4}\left(17\theta_{3}^{4}+\theta_{4}^{4}\right)\right)\right)\right)$
(80)

##  Appendix C Expressions of $f_{p}(\rho)$ for dimers on $2M\times(2N-1)$
cylinder from $p=0$ up to $p=3$.

$\displaystyle f_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{6}\log\left(\frac{2\theta_{2}^{2}}{\theta_{3}\theta_{4}}\right)$
(82) $\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\pi^{3}\rho^{2}\left(\frac{\theta_{4}^{8}}{30}+\frac{1}{30}\theta_{4}^{4}\theta_{2}^{4}-\frac{7\theta_{2}^{8}}{240}\right)$
(83) $\displaystyle f_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{144}\pi^{5}\rho^{3}\left(\frac{4}{3}\rho\left(-\frac{31}{224}\theta_{3}^{2}\theta_{2}^{12}\text{E}-\frac{13}{56}\theta_{3}^{2}\theta_{4}^{4}\theta_{2}^{8}\text{E}+\frac{13}{56}\theta_{4}^{3}\theta_{2}^{8}\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}-\frac{1}{2}\theta_{3}^{2}\theta_{4}\text{E}\right)\right.\right.$
(84) $\displaystyle-$
$\displaystyle\left.\left.\frac{1}{14}\theta_{3}^{2}\theta_{4}^{8}\theta_{2}^{4}\text{E}+\frac{2}{7}\theta_{4}^{7}\theta_{2}^{4}\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}-\frac{1}{2}\theta_{3}^{2}\theta_{4}\text{E}\right)+\frac{2}{7}\theta_{4}^{11}\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}-\frac{1}{2}\theta_{3}^{2}\theta_{4}\text{E}\right)\right)\right.$
(85) $\displaystyle+$
$\displaystyle\left.4\left(\frac{31\theta_{2}^{12}}{1344}+\frac{13}{224}\theta_{4}^{4}\theta_{2}^{8}+\frac{1}{28}\theta_{4}^{8}\theta_{2}^{4}+\frac{\theta_{4}^{12}}{42}\right)\right)$
(86) $\displaystyle f_{3}$ $\displaystyle=$
$\displaystyle\frac{\pi^{7}\rho^{4}}{5760}\left(-\frac{40}{9}\left(-\frac{14}{5}\text{E}\theta_{3}^{2}\theta_{4}^{3}\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}-\frac{1}{2}\text{E}\theta_{3}^{2}\theta_{4}\right)\theta_{2}^{12}+\frac{7}{120}\left(\frac{1}{4}\theta_{3}^{4}\left(12\text{E}\left(\text{E}-\pi\theta_{3}^{2}\right)\right.\right.\right.\right.$
(87) $\displaystyle+$
$\displaystyle\left.\left.\left.\left.\pi^{2}\left(\theta_{2}^{4}+3\theta_{4}^{4}\right)\right)\theta_{4}^{4}+12\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}-\frac{1}{2}\text{E}\theta_{3}^{2}\theta_{4}\right){}^{2}\theta_{4}^{2}\right)\theta_{2}^{12}-\frac{8}{5}\text{E}\theta_{3}^{2}\theta_{4}^{7}\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}\right.\right.\right.$
(88) $\displaystyle-$
$\displaystyle\left.\left.\left.\frac{1}{2}\text{E}\theta_{3}^{2}\theta_{4}\right)\theta_{2}^{8}+\frac{1}{40}\left(\frac{1}{2}\theta_{3}^{4}\left(12\text{E}\left(\text{E}-\pi\theta_{3}^{2}\right)+\pi^{2}\left(\theta_{2}^{4}+3\theta_{4}^{4}\right)\right)\theta_{4}^{8}+56\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}\right.\right.\right.\right.$
(89) $\displaystyle-$
$\displaystyle\left.\left.\left.\left.\frac{1}{2}\text{E}\theta_{3}^{2}\theta_{4}\right){}^{2}\theta_{4}^{6}\right)\theta_{2}^{8}+\frac{16}{5}\text{E}\theta_{3}^{2}\theta_{4}^{11}\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}-\frac{1}{2}\text{E}\theta_{3}^{2}\theta_{4}\right)\theta_{2}^{4}-\frac{1}{15}\left(\frac{3}{4}\theta_{3}^{4}\left(12\text{E}\left(\text{E}-\pi\theta_{3}^{2}\right)\right.\right.\right.\right.$
(90) $\displaystyle+$
$\displaystyle\left.\left.\left.\left.\pi^{2}\left(\theta_{2}^{4}+3\theta_{4}^{4}\right)\right)\theta_{4}^{12}+132\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}-\frac{1}{2}\text{E}\theta_{3}^{2}\theta_{4}\right){}^{2}\theta_{4}^{10}\right)\theta_{2}^{4}+\frac{1}{15}\theta_{4}^{12}\left(\frac{1}{2}\theta_{2}^{4}\theta_{3}^{4}\left(\pi^{2}\theta_{4}^{4}-6\text{E}^{2}\right)\right.\right.\right.$
(91) $\displaystyle-$
$\displaystyle\left.\left.\left.3\text{E}^{2}\theta_{2}^{4}\theta_{3}^{4}\right)+\frac{1}{40}\theta_{4}^{8}\left(14\text{E}^{2}\theta_{2}^{8}\theta_{3}^{4}-\theta_{2}^{8}\theta_{3}^{4}\left(\pi^{2}\theta_{4}^{4}-6\text{E}^{2}\right)\right)+\frac{7}{120}\theta_{4}^{4}\left(33\text{E}^{2}\theta_{2}^{12}\theta_{3}^{4}\right.\right.\right.$
(92) $\displaystyle-$
$\displaystyle\left.\left.\left.\frac{3}{2}\theta_{2}^{12}\theta_{3}^{4}\left(\pi^{2}\theta_{4}^{4}-6\text{E}^{2}\right)\right)+\frac{127\left(60\text{E}^{2}\theta_{2}^{16}\theta_{3}^{4}-2\theta_{2}^{16}\theta_{3}^{4}\left(\pi^{2}\theta_{4}^{4}-6\text{E}^{2}\right)\right)}{3840}+\frac{1}{30}\left(-\theta_{3}^{4}\left(12\text{E}\left(\text{E}\right.\right.\right.\right.\right.$
(93) $\displaystyle-$
$\displaystyle\left.\left.\left.\left.\left.\pi\theta_{3}^{2}\right)+\pi^{2}\left(\theta_{2}^{4}+3\theta_{4}^{4}\right)\right)\theta_{4}^{16}-240\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}-\frac{1}{2}\text{E}\theta_{3}^{2}\theta_{4}\right){}^{2}\theta_{4}^{14}\right)\right)\rho^{2}\right.$
(94) $\displaystyle-$
$\displaystyle\left.40\left(-\frac{127}{480}\text{E}\theta_{3}^{2}\theta_{2}^{16}-\frac{7}{20}\text{E}\theta_{3}^{2}\theta_{4}^{4}\theta_{2}^{12}+\frac{7}{30}\theta_{4}^{3}\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}-\frac{1}{2}\text{E}\theta_{3}^{2}\theta_{4}\right)\theta_{2}^{12}-\frac{1}{10}\text{E}\theta_{3}^{2}\theta_{4}^{8}\theta_{2}^{8}\right.\right.$
(95) $\displaystyle+$
$\displaystyle\left.\left.\frac{1}{5}\theta_{4}^{7}\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}-\frac{1}{2}\text{E}\theta_{3}^{2}\theta_{4}\right)\theta_{2}^{8}+\frac{2}{15}\text{E}\theta_{3}^{2}\theta_{4}^{12}\theta_{2}^{4}-\frac{4}{5}\theta_{4}^{11}\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}-\frac{1}{2}\text{E}\theta_{3}^{2}\theta_{4}\right)\theta_{2}^{4}\right.\right.$
(96) $\displaystyle-$
$\displaystyle\left.\left.\frac{8}{15}\theta_{4}^{15}\left(\frac{1}{4}\pi\theta_{3}^{4}\theta_{4}-\frac{1}{2}\text{E}\theta_{3}^{2}\theta_{4}\right)\right)\rho-\frac{632}{7}\left(\frac{127\theta_{2}^{16}}{3840}+\frac{7}{120}\theta_{4}^{4}\theta_{2}^{12}+\frac{1}{40}\theta_{4}^{8}\theta_{2}^{8}\right.\right.$
(97) $\displaystyle-$
$\displaystyle\left.\left.\frac{1}{15}\theta_{4}^{12}\theta_{2}^{4}-\frac{\theta_{4}^{16}}{30}\right)\right)$
(98)

##  Appendix D Relation between elliptic theta functions, elliptic integral
and gamma function

The elliptic theta functions and the elliptic integral of the second kind $E$
at particular values of the aspect ratios $\rho=1/2,1,2$ and $4$ given by

$\displaystyle\theta_{2}=\frac{2^{7/8}\sqrt[8]{x}}{\sqrt[4]{\pi}},\hskip
11.38092pt\theta_{3}=\frac{\sqrt{2+\sqrt{2}}\sqrt[8]{x}}{\sqrt[4]{\pi}},\hskip
11.38092pt\theta_{4}=\frac{\sqrt{2-\sqrt{2}}\sqrt[8]{x}}{\sqrt[4]{\pi}},$
$\displaystyle
E=\frac{\sqrt{\pi}\sqrt[4]{x}}{2+\sqrt{2}}+\frac{\sqrt{\pi}}{\left(2+\sqrt{2}\right)\sqrt[4]{x}}$
(100)

for $\rho=1/2$,

$\displaystyle\theta_{2}=\theta_{4}=\sqrt[4]{\frac{2}{\pi}}\sqrt[8]{x},\hskip
11.38092pt\theta_{3}=\frac{\sqrt{2}}{\sqrt[4]{\pi}}\sqrt[8]{x},\hskip
11.38092ptE=\frac{1}{4}\sqrt{\pi}\sqrt[4]{\frac{1}{x}}+\frac{\sqrt{\pi}}{2\sqrt[4]{\frac{1}{x}}}$
(101)

for $\rho=1$,

$\displaystyle\theta_{2}=\frac{\sqrt{1-\frac{1}{\sqrt{2}}}\sqrt[8]{x}}{\sqrt[4]{\pi}},\hskip
11.38092pt\theta_{3}=\frac{\sqrt{1+\frac{1}{\sqrt{2}}}\sqrt[8]{x}}{\sqrt[4]{\pi}},\hskip
11.38092pt\theta_{4}=\frac{2^{3/8}\sqrt[8]{x}}{\sqrt[4]{\pi}},$ $\displaystyle
E=\frac{\sqrt{\pi}\left(\sqrt{2}\sqrt{x}+\frac{1}{2+\sqrt{2}}\right)}{2\sqrt[4]{x}}$
(102)

for $\rho=2$,

$\displaystyle\theta_{2}=\frac{\left(\sqrt[4]{2}-1\right)\sqrt[8]{x}}{2^{3/4}\sqrt[4]{\pi}},\hskip
11.38092pt\theta_{3}=\frac{\left(1+\sqrt[4]{2}\right)\sqrt[8]{x}}{2^{3/4}\sqrt[4]{\pi}},\hskip
11.38092pt\theta_{4}=\frac{\sqrt[4]{\frac{2+\sqrt{2}}{\pi}}\sqrt[8]{x}}{\sqrt[16]{2}},$
$\displaystyle E=\frac{\sqrt{\pi}\left(\left(62+54\sqrt[4]{2}+42\sqrt{2}+34\
2^{3/4}\right)\sqrt{x}+4\
2^{3/4}+6\sqrt{2}+4\sqrt[4]{2}+3\right)}{2\left(1+\sqrt[4]{2}\right)^{4}\left(2+\sqrt{2}+2\
2^{3/4}\right)\sqrt[4]{x}}$ (103)

for $\rho=4$, where

$\displaystyle
x=\frac{\pi^{4}}{16\Gamma\left(\frac{3}{4}\right)^{8}}=\frac{\Gamma[1/4]^{8}}{256\pi^{4}}=1.197316987373152280414040\ldots$

and $\Gamma(z)$ is the gamma function.

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MIDLMedical Imaging with Deep Learning 2019 MIDL 2019 – Extended Abstract
Track Richard Du1<EMAIL_ADDRESS>
1 Department of Diagnostic Radiology, Li Ka Shing Faculty of Medicine, The
University of Hong Kong, Hong Kong SAR and Peng Cao1<EMAIL_ADDRESS>and Lujun
Han2<EMAIL_ADDRESS>
2 Sun Yat-Sen University Cancer Center, Guangzhou, China and Qiyong Ai3
<EMAIL_ADDRESS>
3 Department of Imaging and Interventional Radiology, Faculty of Medicine, The
Chinese University of Hong Kong, Hong Kong SAR and Ann D.
<EMAIL_ADDRESS>and Varut Vardhanabhuti1<EMAIL_ADDRESS>
# Deep convolution neural network model for automatic risk assessment of
patients with non-metastatic nasopharyngeal carcinoma

##  1 Introduction

Nasopharyngeal Carcinoma (NPC) is a malignant tumour that develops in the
nasopharynx. NPC is endemic in the south-east Asia with approximately a third
of the world incidences (Tang et al., 2016). With the advent of intensity-
modulated radiotherapy, better sparing of adjacent organs at risk and
excellent locoregional control are being achieved. Consequently, this had led
to pretreatment clinical staging classification to be less prognostic of
outcomes such as recurrence after treatment (Liu et al., 2017). Alternative
pretreatment strategies for prognosis of NPC after treatment are needed to
provide better risk stratification for NPC. In this study, we attempt to
develop an automatic deep convolution neural network model to predict 3-year
disease progression.

##  2 Methods

###  2.1 Data set

596 non-metastatic NPC patients were retrospectively obtained from four
independent centres in Hong Kong and China. Head and neck contrast enhanced
T1-weighted (T1C) and T2-weighted (T2) MRI scan along with routine
pretreatment TNM staging classification and 3-year progression-free survival
status were retrieved for the study. The 3-year disease progression rate was
found to be between 10-20% across the cohorts. All scans were resampled to
1x1x6cm resolution. For development of the model, 450 patients from three
centres were allocated for the training set, and 146 from one other centre was
allocated for the testing set. The primary NPC tumours of a subset of the 285
patients from training set and 30 patients from testing set were manually
delineated by a board-certified radiologist. All manual segmentation was
performed on the T1C scans.

###  2.2 Proposed model and training

The proposed model consists of two separate networks which perform
segmentation of the primary NPC tumour and then uses segmentation for
classification of 3-year disease progression. For segmentation, inspired by
the VNet architecture (Milletari et al., 2016), we modified the architecture
to provide two separate encoding routes for encoding the T1C and T2 scans. The
resulting encoded latent features from the two encoders are combined for the
segmentation decoder. Additionally, a classification encoder was added to
classify the overall and T stage of the tumour. The intuition behind the
classification route is to force the network to encode features related to the
tumour. The proposed network architecture, named as V2NetCls, are shown in
Figure LABEL:fig:WNET. For training, 15% of the subset of patients with
segmentation was randomly split for validation (n = 43) and the rest used for
training (n = 243). Adam optimiser with a learning rate of 1e-4 and batch of 2
was used. Dice loss was used as it known to be superior in class imbalanced
situation Milletari et al. (2016). For baseline, a VNet trained with T1C scan
only and the V2Net without classification was also trained. Once the
segmentation network is trained, the output was used to crop a 64x64x12
bounding box of the segmented tumour on T1C and T2 scan for classification of
3-year PFS. ResNet18 was used as the network to classifier 3-year disease
progression He et al. (2015). The input to the network is a 64x64x12 volume
with three channels corresponding to the T1C, T2 and the segmentation mask.
Similarly, an 85/15 split was used for training (n = 384) and validation (n =
68). Weighted binary cross-entropy loss was used for training.

##  3 Results

The segmentation performance of the proposed and baseline networks is given in
Table LABEL:tab:example. The proposed V2NetCls achieved the best performance
compared to the two baseline networks, hence was subsequently used for the
classification of 3-year disease progression. The classification network
achieved an AUC = 0.828 (sensitivity = 0.833; specificity=0.887) in the
validation set but was unable to generalise to the testing set (AUC = 0.689;
sensitivity = 0.685; specificity = 0.716)

tab:example  Model Data set Median Dice (IQR) VNet T1C Validation set (n = 43)
0.558 (0.397 - 0.646) Testing set (n = 30) 0.437 (0.329 - 0.565) V2Net T1C+T2
Validation set (n = 43) 0.589 (0.455 - 0.675) Testing set (n = 30) 0.559
(0.429 - 0.656) V2NetCls T1C+T2 Validation set (n = 43) 0.612 (0.456 - 0.693)
Testing set (n = 30) 0.585 (0.480 - 0.681)

Table 1: Segmentation performance of the experimented networks. IQR,
interquartile-range

##  4 Discussion

Despite the low multicentre performance, our preliminary results show that
deep learning may offer prognostication of disease progression of NPC patients
after treatment. Despite sub-par segmentation performance, the segmentation
was able to localise the tumour and act as prior knowledge for the
classification. More samples number may be needed to increase the segmentation
performance as seen in Lin et al. (2019). The classification performance was
comparable to a previous NPC model based on hand-crafted radiomics features
which achieved an AUC = 0.78 in a internal test set Zhang et al. (2017). One
advantage of our model is that it does not require manual segmentation of the
region of interest, hence reducing clinician’s burden. However further
development in generalising multicentre data set are needed before clinical
application of deep learning models in assessment of NPC.

fig:WNET

Figure 1: Architecture of the proposed V2NetCls for segmentation of primary
NPC tumour

RD is supported by a postgraduate fellowship from the Lee Shau Kee Foundation.

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