Large-diffusion asymptotics for voter-model Lyapunov exponents

Let $d\geq5$, let $\rho\in(0,\infty)$ and $p\in\mathbb{N}$, and assume the symmetric random-walk condition denoted by (SRW) in the source. Let $p_t(0,0)$ be the return transition probability, and define

$$G_d=\int_0^\infty p_t(0,0)\,dt,\qquad G_d^\ast=\int_0^\infty t\,p_t(0,0)\,dt.$$

Also define

$$\mathcal{P}_5=\sup_{\substack{f\in H^1(\mathbb{R}^5)\\\|f\|_2=1}}\left[\int_{\mathbb{R}^5}dx\,|f(x)|^2\int_{\mathbb{R}^5}dy\,|f(y)|^2\frac{1}{16\pi^2\|x-y\|}-\int_{\mathbb{R}^5}dx\,|\nabla f(x)|^2\right].$$

Large-diffusion asymptotic conjecture. The limit

$$\lim_{\kappa\to\infty}2d\kappa[\lambda_p(\kappa)-\rho\gamma]=\rho(1-\rho)\gamma^2\frac{G_d^\ast}{G_d}+1_{\{d=5\}}(2d)^3\left[\rho(1-\rho)\gamma^2\frac{p}{G_d}\right]^2\mathcal{P}_5$$

exists.

This conjecture specifies the first large-$\kappa$ correction to the Lyapunov exponent in dimensions at least five, including an additional dimension-five term. The source does not provide a resolution.