Scaling-limit conjecture for directed polymers on fractal graphs

Let the underlying random walk satisfy the local limit theorem in the sense discussed by Croydon and Hambly. For $T>0$, let $Z^0_{Tn,\beta_n}$ be the directed-polymer partition function, let $Q$ denote the disorder expectation, and set

$$\beta_n=n^{-\frac{d_w-d_f}{2d_w}}.$$

Let $\bar{V}$ be the limiting fractal space with measure $\mu$, and define

$$\mathcal{Z}_T=\int_{\bar{V}}\mathcal{Z}_{T,x}\,\mu(dx).$$

Here $\mathcal{Z}_{t,x}$ is specified by

$$\mathcal{Z}_{t,x}=p_t(x)+\hat\beta\int_0^t\int_{\bar{V}}\left(\int_{\bar{V}}p_{t-s}(x-z)\mathcal{Z}_{s,z-y}\,\mu(dz)\right)\mathcal{W}(ds,dy),$$

where $\hat\beta$ is a constant depending on the underlying random walk, $p_t(x)$ is the heat kernel of Brownian motion on the fractal graph, and $\mathcal{W}$ is white noise on $(\bar{V},\mu)$. Scaling-limit conjecture. For each $T>0$,

$$\frac{Z^0_{Tn,\beta_n}}{Q\left[Z^0_{Tn,\beta_n}\right]}\Rightarrow\mathcal{Z}_T.$$

This predicts convergence of the normalized polymer partition function to a stochastic heat-equation-type limit on the fractal scaling space. Simple random walk on the Sierpinski gasket graph satisfies the relevant local limit theorem, while existence and uniqueness for related stochastic partial differential equations on measure spaces have been studied; the asserted convergence in this general setting remains open.