Rate of convergence of the critical point of the memory-$\tau$ self-avoiding walk in dimensions $d>4$

We consider spread-out models of the self-avoiding walk and its finite-memory version, known as the memory-$\tau$ walk, which prohibits loops whose length is at most $\tau$, in dimensions $d>4$. The critical point is defined as the radius of convergence of the generating function for each model. It is known that the critical point of the memory-$\tau$ walk is non-decreasing in $\tau$ and converges to that of the self-avoiding walk as $\tau$ tends to infinity. In this paper, we study the rate at which the critical point of the memory-$\tau$ walk converges to that of the self-avoiding walk and show that the order is ${\tau }^{-(d-2)\mathrm{/}2}$. The proof relies on the lace expansion, introduced by Brydges and Spencer.