  How long does a self-avoiding walk on a discrete $d$-dimensional torus have
to be before it begins to behave differently from a self-avoiding walk on
$\mathbb{Z}^d$? We consider a version of this question for weakly self-avoiding
walk on a torus in dimensions $d>4$. On $\mathbb{Z}^d$ for $d>4$, the partition
function for $n$-step weakly self-avoiding walk is known to be asymptotically
purely exponential, of the form $A\mu^n$, where $\mu$ is the growth constant
for weakly self-avoiding walk on $\mathbb{Z}^d$. We prove the identical
asymptotic behaviour $A\mu^n$ on the torus (with the same $A$ and $\mu$ as on
$\mathbb{Z}^d$) until $n$ reaches order $V^{1/2}$, where $V$ is the number of
vertices in the torus. This shows that the walk must have length of order at
least $V^{1/2}$ before it "feels" the torus in its leading asymptotics. Our
results support the conjecture that the behaviour of the partition function
does change once $n$ reaches $V^{1/2}$, and we relate this to a conjectural
critical scaling window which separates the dilute phase $n \ll V^{1/2}$ from
the dense phase $n \gg V^{1/2}$. 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.
