How long does a self-avoiding walk on a discrete
-dimensional
torus have to be before it begins to behave differently from a self-avoiding walk on
? We
consider a version of this question for weakly self-avoiding walk on a torus in dimensions
. On
for
, the partition
function for
-step
weakly self-avoiding walk is known to be asymptotically purely exponential, of the
form
,
where
is the growth constant for weakly self-avoiding walk on
. We prove the identical
asymptotic behaviour
on
the torus (with the same
and
as
on
) until
reaches
order
,
where
is
the number of vertices in the torus. This shows that the walk must have length of order
at least
before it “feels” the torus in its leading asymptotics. Our results support the
conjecture that the behaviour of the partition function does change once
reaches
, and we
relate this to a conjectural critical scaling window which separates the dilute phase
from the
dense phase
.
To prove the conjecture and to establish the existence of the scaling window remains
a challenging open problem. The proof uses a novel lace expansion analysis
based on the “plateau” for the torus two-point function obtained in previous
work.