<s>
In	O
statistical	O
mechanics	O
and	O
mathematics	O
,	O
the	O
Bethe	B-Algorithm
lattice	I-Algorithm
(	O
also	O
called	O
a	O
regular	O
tree	O
)	O
is	O
an	O
infinite	O
connected	O
cycle-free	O
graph	O
where	O
all	O
vertices	O
have	O
the	O
same	O
number	O
of	O
neighbors	O
.	O
</s>
<s>
The	O
Bethe	B-Algorithm
lattice	I-Algorithm
was	O
introduced	O
into	O
the	O
physics	O
literature	O
by	O
Hans	O
Bethe	O
in	O
1935	O
.	O
</s>
<s>
Due	O
to	O
its	O
distinctive	O
topological	O
structure	O
,	O
the	O
statistical	O
mechanics	O
of	O
lattice	B-Algorithm
models	I-Algorithm
on	O
this	O
graph	O
are	O
often	O
easier	O
to	O
solve	O
than	O
on	O
other	O
lattices	O
.	O
</s>
<s>
When	O
working	O
with	O
the	O
Bethe	B-Algorithm
lattice	I-Algorithm
,	O
it	O
is	O
often	O
convenient	O
to	O
mark	O
a	O
given	O
vertex	O
as	O
the	O
root	O
,	O
to	O
be	O
used	O
as	O
a	O
reference	O
point	O
when	O
considering	O
local	O
properties	O
of	O
the	O
graph	O
.	O
</s>
<s>
The	O
Bethe	B-Algorithm
lattice	I-Algorithm
is	O
of	O
interest	O
in	O
statistical	O
mechanics	O
mainly	O
because	O
lattice	B-Algorithm
models	I-Algorithm
on	O
the	O
Bethe	B-Algorithm
lattice	I-Algorithm
are	O
often	O
easier	O
to	O
solve	O
than	O
on	O
other	O
lattices	O
,	O
such	O
as	O
the	O
two-dimensional	O
square	O
lattice	O
.	O
</s>
<s>
While	O
the	O
Bethe	B-Algorithm
lattice	I-Algorithm
does	O
not	O
as	O
closely	O
approximate	O
the	O
interactions	O
in	O
physical	O
materials	O
as	O
other	O
lattices	O
,	O
it	O
can	O
still	O
provide	O
useful	O
insight	O
.	O
</s>
<s>
In	O
the	O
case	O
when	O
the	O
system	O
is	O
ferromagnetic	O
,	O
the	O
above	O
sequence	O
converges	O
,	O
so	O
we	O
may	O
take	O
the	O
limit	O
to	O
evaluate	O
the	O
magnetization	O
on	O
the	O
Bethe	B-Algorithm
lattice	I-Algorithm
.	O
</s>
<s>
The	O
probability	O
that	O
a	O
random	O
walk	O
on	O
a	O
Bethe	B-Algorithm
lattice	I-Algorithm
of	O
degree	O
starting	O
at	O
a	O
given	O
vertex	O
eventually	O
returns	O
to	O
that	O
vertex	O
is	O
given	O
by	O
.	O
</s>
<s>
Such	O
a	O
lattice	O
is	O
4-regular	O
,	O
but	O
the	O
4-regular	O
Bethe	B-Algorithm
lattice	I-Algorithm
has	O
a	O
return	O
probability	O
of	O
1/3	O
.	O
</s>
<s>
One	O
can	O
easily	O
bound	O
the	O
number	O
of	O
closed	O
walks	O
of	O
length	O
starting	O
at	O
a	O
given	O
vertex	O
of	O
the	O
Bethe	B-Algorithm
Lattice	I-Algorithm
with	O
degree	O
from	O
below	O
.	O
</s>
<s>
Let	O
be	O
a	O
-regular	O
graph	O
with	O
vertices	O
,	O
and	O
let	O
be	O
its	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
number	O
of	O
closed	O
walks	O
on	O
is	O
at	O
least	O
times	O
the	O
number	O
of	O
closed	O
walks	O
on	O
the	O
Bethe	B-Algorithm
lattice	I-Algorithm
with	O
degree	O
starting	O
at	O
a	O
particular	O
vertex	O
,	O
as	O
we	O
can	O
map	O
the	O
walks	O
on	O
the	O
Bethe	B-Algorithm
lattice	I-Algorithm
to	O
the	O
walks	O
on	O
that	O
start	O
at	O
a	O
given	O
vertex	O
and	O
only	O
go	O
back	O
on	O
paths	O
that	O
were	O
already	O
tread	O
.	O
</s>
<s>
Bethe	B-Algorithm
lattices	I-Algorithm
also	O
occur	O
as	O
the	O
discrete	O
subgroups	O
of	O
certain	O
hyperbolic	O
Lie	O
groups	O
,	O
such	O
as	O
the	O
Fuchsian	O
groups	O
.	O
</s>
