I have come to suspect that the following is true (and have confirmed it with some numerical experiments) but I have no idea how to prove it.
Background: Let $f(z) = \sum_{n=0}^N a_n z^n$ be some polynomial of order $N$ (with $a_n \in \mathbb R$) where for convenience we take $a_N = 1$. This defines an associated recursion relation: $$c_{k+N} = - \sum_{n=0}^{N-1} a_n \; c_{k+n} \qquad (k \in \mathbb N)$$ Each zero $\lambda_i$ of $f$ (where we can take them all to be distinct, and the index '$i$' labels from small to large) defines a linearly independent solution, i.e. $c_k^{(i)} := \lambda_i^k$. The asymptotics of this series is trivially $$c_k^{(i)} \sim |\lambda_i|^k$$
Possible conjecture: Suppose we now add some 'noise' to the coefficients, i.e. consider the recursion relation with variable coefficients: $$c_{k+N} = - \sum_{n=0}^{N-1} a_{n,k} \; c_{k+n} \qquad (k \in \mathbb N)$$ where $a_{n,k} = a_n + \varepsilon_{n,k}$ (but still $a_N = 1$). For each $n$ and $k$, we let $\varepsilon_{n,k}$ be a random variable out of the flat distribution on $[-W,W]$ (where $W \in \mathbb R^+$ is some number we fix). Note that for any $k$ we can define the perturbed polynomial $f_k(z) = \sum_{n=0}^N a_{n,k} z^n$, which will have some perturbed zeros $\lambda_{i}(\varepsilon)$. Suppose we take $W$ small enough such that the $N$ zeros 'stay away from one another', more precisely: for any possible realization of $\varepsilon_{n,k}$, the distance $|\lambda_{i}(\varepsilon) - \lambda_j|$ is minimal for $j= i$. Then I conjecture that there are still $N$ linearly independent solutions $\{ (c_0^{(i)},c_1^{(i)},c_2^{(i)},\cdots)\}_{i=1,2,\cdots,N}$ (this part is not hard to see), and each has the asymptotics $$ c_k^{(i)} \sim \left(\int_{-W}^W |\lambda_i(\varepsilon)| \; P(\varepsilon) \mathrm d \epsilon\right)^k$$ (where of course the probability distribution is $P(\varepsilon) = \frac{1}{2W}$).
Note that the latter statement (about the asymptotics) should be a probabilistic statement, i.e. that it is the most likely outcome, but I am not quite sure how (strongly) to formalize that aspect.
