Asymptotics of recurrence relation coefficients, Hankel determinant ratios, and root products associated with Laurent polynomials orthogonal with respect to varying exponential weights

Let Lambda(R) denote the linear space over R spanned by z(k), k is an element of Z. Define the real inner product <center dot, center dot >(L) : Lambda(R) x Lambda(R) -> R, (f, g) bar right arrow integral(R) f(s)g(s)exp(-NV(s))ds, N is an element of N, where V satisfies: (i) V is real analytic on R \ {0}; (ii) lim(vertical bar x vertical bar ->infinity) (V(x)/ln(x(2) + 1)) = +infinity; and (iii) lim(vertical bar x vertical bar ->infinity) (V(x)/ln(x(-2) + 1)) = +infinity. Orthogonalisation of the (ordered) base {1, z(-1), z, z(-2), z(2),..., z(-k), z(k),...} with respect to <center dot, center dot >(L) yields the even degree and odd degree orthonormal Laurent polynomials (OLPs) {phi(m)(x)}(m=0)(infinity): phi(2n)(z) = (n)Sigma(k=-n) xi((2n))(k)z(k), xi((2n))(n) > 0, and phi(2n+1)(z) = (n)Sigma(k=-n-1) xi((2n+1))(k)z(k), xi((2n))(-n-1) > 0. Associated with the even degree and odd degree OLPs are the following two pairs of recurrence relations: z phi(2n)(z) = c(2n)(#)phi(2n-2)(z) + b(2n)(#)phi(2n-1)(z) + a(2n)(#)phi(2n)(z) + b(2n+1)(#)phi(2n+1)(z) + c(2n+2)(#)phi(2n+2)(z) and z phi(2n+1)(z) = b(2n+1)(#)phi(2n)(z) + a(2n+1)(#)phi(2n+1)(z) + b(2n+2)(#)phi(2n+2)(z), where c(0)(#) = b(0)(#), and c(2k)(#) > 0, k is an element of N, and z(-1)phi(2n+1)(z) = gamma(#)(2n+1)phi(2n-1)(z) + beta(#)(2n+1)phi(2n)(z) + alpha(#)(2n+1)phi(2n+1)(z) + beta(#)(2n+2)phi(2n+2)(z) + gamma(#)(2n+3)phi(2n+3)(z) and z(-1)phi(2n)(z) = beta(#)(2n)phi(2n-1)(z) + alpha(#)(2n)phi(2n)(z) + beta(#)(2n+1)phi(2n+1)(z), where beta(#)(0) = gamma(#)(1) = 0, beta(#)(1) > 0, and gamma(#)(2l+1) > 0, l is an element of N. Asymptotics in the double-scaling limit N, n -> infinity such that N/n = 1+o(1) of the coefficients of these two pairs of recurrence relations, Hankel determinant ratios associated with the real-valued, bi-infinite strong moment sequence {c(k) = integral(R)s(k) exp(-NV(s))ds} (k is an element of Z), and the products of the (real) roots of the OLPs are obtained by formulating the even degree and odd degree OLP problems as matrix Riemann-Hilbert problems on R, and then extracting the large-n behaviours by applying the non-linear steepest-descent method introduced in (Ann. Math. 137(2): 295-368, 1993) and further developed in (Commun. Pure Appl. Math. 48(3): 277-337, 1995) and (Int. Math. Res. Not. 6: 285-299, 1997).