=\nu
d=D_{\mathrm{eff}}$, where $d$ is a dimension of the underlying
lattice. Since the value $1/\nu$ for both the heptagonal and dual
heptagonal lattices are smaller than $0.5$, our results can be
explained by an effective dimensionality of the two hyperbolic
lattices greater than $2$. (This is reasonable---the
dimensionality of hyperbolic surfaces, embedded in an Euclidean
geometry, is also  larger than two.)

\subsection{Effect of boundary nodes}
From our numerical results, we know that the critical behaviors of
the majority-vote model on the heptagonal and dual heptagonal
lattices are different. One potential explanation comes from by
the different topological structure of the boundary nodes. (The
boundary nodes of the regular heptagonal lattice interacts less
with the inner part of the surface compared with the dual
heptagonal case.)

To explore this boundary effect, one straightforward way is to
consider a (regular or dual) heptagonal lattice with a large size,
and investigate the magnetization in the center and compare that
with a heptagonal lattice of a smaller size. In particular, we
perform simulations on the heptagonal lattice at level $7$
($N=5887$), and track the magnetization in the inner part within
level $3$, $4$, $5$, and $6$ separately. Then we compare the
results with the magnetization of Fig.~\ref{MVH}(a). The same
procedure is also done for dual heptagonal lattice at level $8$
($N=4264$), but within level $4$, $5$, $6$, and $7$ respectively.
The numerical results are summarized in Figs.~\ref{innerh}
and~\ref{innerdh}. We note that the nonvanishing boundary nodes
play different roles in the ordering processes. As can be observed
in Fig.~\ref{innerh}, the magnetization in all the inner parts of
a big heptagonal lattice are greater than that on a heptagonal
lattice with same size, which indicates that the existence of a
higher level of boundary can facilitate the ordering of the inner
spins. This picture is changed, however, on the dual heptagonal
lattice. In the subcritical region $q<q_c$, the boundary nodes
promote the ordering process as before. In the supercritical
region $q>q_c$, on the contrary, they impose opposite influence on
the ordering process driving the system towards more disorder
[Fig.~\ref{innerdh}]. A structural cause for this phenomenon is,
we believe, the different local topologies of the two lattices.
Looking at Fig.~\ref{HL}, the nodes one step from the boundary
have fewer connections to the boundary in the heptagonal lattice
than its dual (the path-length going from the periphery to the
center, for systems of the same size, are longer). The boundary is
also more indirectly coupled to the interior in the heptagonal
lattice.  The boundary nodes of the dual heptagonal lattice should
thus have a stronger influence on the configuration of the
interior (we hesitate to say ``bulk properties'' since the
boundary is a finite fraction of the interior). We expect similar
stronger boundary effects for dual heptagonal lattices also exists
for other statistical spin models defined.

\begin{figure}[h]
\includegraphics[width=0.9\linewidth]{innera.ps}
\caption{(Color online) Magnetization $M$ as a function of the
noise parameter $q$. (a) Filled squares represent the
magnetization at the inner level $3$ of the heptagonal lattice
with  $7$ levels in total. Open circles represent the heptagonal
lattice with $3$ levels. (b), (c) and (d) show the same situation as (a) but
for different levels.}\label{innerh}
\end{figure}

\begin{figure}[h]
\includegraphics[width=0.9\linewidth]{innerb.ps}
\caption{(Color online) Magnetization $M$ as a function of the
noise parameter $q$. (a) Filled squares symbolize the
magnetization in the inner level $4$ of the dual heptagonal
lattice with $8$ levels in total. Open circles represent the
heptagonal lattice with $4$ levels. (b), (c) and (d) show  plots
corresponding to (a) for other levels.}\label{innerdh}
\end{figure}

\subsection{Comparisons with the Ising model}
In Ref.~\cite{Oliveira1992jsp}, Oliveira found that the critical
exponents for the majority-vote model on square lattice with
periodic boundary condition are $\nu=0.99\pm0.05$,
$\gamma/\nu=1.73\pm0.05$, and $\beta/\nu=0.125\pm0.005$. These
results demonstrate the majority-vote model on a torus has the
same universal critical behavior as the equilibrium Ising model.
Recently, Yang \emph{et al.}~investigated a slightly different
majority-vote model on \emph{d}-dimensional hypercubic lattices.
Their model is identical to the original one by simple replacement
of the temperature parameter to a noise parameter like in the
present paper, $\tanh(1/kT)=(1-2q)$~\cite{Yang2008pre}. Their
simulation results suggest that the critical exponents for the
majority-vote model in three dimensions are different from those
of the Ising model, and that the results for four and five
dimensions are far from the standard mean-field values. For their
version of the majority-vote model in two dimensions, the global
and local configuration energy differences between before and
after spin-flipping are not identical~\cite{Note}, but the sign of
both energies is the same, whereas for $d\geq3$, the sign of the
two types of energy differences is not always the same. For the
Ising model, however, the differences of the global and local
configuration energy are exactly identical, regardless of
dimension. They conjectured that the discordance of the sign of
the energy difference between the global and local energy is
responsible for the different critical behaviors. Observe that our
critical exponents $1/\nu\approx0.3$ and $0.33$ in
Fig.~\ref{Hcollapse}(a) and Fig.~\ref{DHcollapse}(a), which are
smaller than $0.5$,  so from the hyperscaling relation $\nu d=1$
($d$ is the dimension) we see that  our hyperbolic lattices can be
described as having an effective dimensionality greater than $2$,
but smaller than $4$. In this sense, our present work provide
complementary support for the results obtained in
Ref.~\cite{Yang2008pre}. In another recent
work~\cite{Shima2006jpa}, Shima~\emph{et al.} studied the Ising
model on heptagonal lattices, and found the critical exponent
$\gamma/\nu=0.655$. Since our critical exponent $\gamma/\nu$, also
considering the estimated error, is quite different from this
value, we believe that the majority-vote model on the heptagonal
lattice belongs to a different universality class as the Ising
model on this topology. This result can be regarded as another
evidence for the conjecture by Yang~\textit{et
al.}~\cite{Yang2008pre} mentioned above.

\section{Conclusions}\label{conclusion}
To summarize, we have studied the critical behavior of the
majority-vote model on the heptagonal and dual heptagonal
lattices. These lattices possess a peculiar property: the ratio of
the size of the boundary to the total size  remains finite even in
the thermodynamic limit. Finite-size scaling analysis reveals that
the critical exponents for magnetization and susceptibility
deviate from those of the majority-vote model on a torus (which
belongs to the same universality class as the equilibrium Ising
model) and are also different from those of the Ising model on
heptagonal and dual heptagonal lattices. In particular, the best
fit of these exponents provided $\beta/\nu=0.114(5)$,
$\gamma/\nu=0.721(7)$ for the heptagonal lattice, and
$\beta/\nu=0.093(6)$, $\gamma/\nu=0.761(9)$ for the dual
heptagonal lattice. By comparing to the majority-vote model on
square lattices with free boundary conditions, we found that the
free boundaries result in strong finite-size scaling effect, which
in turn leads to the measured effective dimensionality smaller
than unity. Nonetheless, we believe that the critical exponents of
the majority-vote model defined on negatively curved surface also
satisfy the hyperscaling relation
$2\beta/\nu+\gamma/\nu=D_{\mathrm{eff}}$.

Furthermore, for the hyperbolic lattices, we also investigated the
effect of the boundary nodes on the ordering process. It was shown
that the boundary nodes have different functions in our two
lattices. For the heptagonal lattice, the boundary has a positive