The scaling conjecture for Lyapunov exponents of the voter model on a catalyst

Let $p_t(x,y)$ be the transition probability at time $t$ for the rate-one random walk with kernel $p(,)$, and let $\blambda_p(\kappa)$ denote the associated Lyapunov exponent. Suppose that $p(,)$ is a simple random walk. 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

$$\mathcal{P}_5=\sup_{f\in H^1(\mathbb{R}^5),\ \|f\|_2=1}\left[\int_{\mathbb{R}^5}\int_{\mathbb{R}^5}\frac{f^2(x)f^2(y)}{16\pi^2\|x-y\|}\,dx\,dy-\|\nabla f\|_2^2\right]\in(0,\infty),$$

where $H^1(\mathbb{R}^5)=\{f:\mathbb{R}^5\to\mathbb{R}:f,\nabla f\in L^2(\mathbb{R}^5)\}$. The scaling conjecture. For all $d\geq5$, $p\in\mathbb{N}$, $\gamma\in(0,\infty)$ and $\rho\in(0,1)$,

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

This refines the stated asymptotic result for the Lyapunov exponent and predicts the dimension-five correction through the variational constant $\mathcal{P}_5$.