Evaluations of Tutte polynomials of regular graphs

Let $T_G(x,y)$ be the Tutte polynomial of a graph $G$. In this paper we show that if $(G_n{)}_n$ is a sequence of $d$-regular graphs with girth $g(G_n)\to \infty$,thenfor$x\geq 1$and$0\leq y\leq 1$wehave$$\lim_{n\to \infty}T_{G_n}(x,y)^{1/v(G_n)}=t_d(x,y),$$where$$t_d(x,y)=\left{\begin{array}{lc} (d-1)\left(\frac{(d-1)^2}{(d-1)^2-x}\right)^{d/2-1}&\ \ \mbox{if}\ x\leq d-1,\ x\left(1+\frac{1}{x-1}\right)^{d/2-1} &\ \ \mbox{if}\ x> d-1. \end{array}\right.$$independentlyof$y$if$0\leq y\leq 1$\mathrm{.}If$(G_n)_n$ is a sequence of random $d$-regular graphs, then the same statement holds true asymptotically almost surely. This theorem generalizes results of McKay ($x=1,y=1$, spanning trees of random $d$-regular graphs) and Lyons ($x=1,y=1$, spanning trees of large-girth $d$-regular graphs). Interesting special cases are $T_G(2,1)$ counting the number of spanning forests, $T_G(2,0)$ counting the number of acyclic orientations.