Given a polynomial $$P(z)=\sum_{n=0}^N a_n z^n$$ with real coefficients distributed as a gaussian curve $a_n=\frac{1}{\sigma\sqrt{2\pi}}e^{(n-b)^2/2\sigma}$ ($b>0$). The sum of all the polynomial coefficients have to be equal to $1$ (I guess the normalization is not the main point of the problem). Now, let's suppose that a gap is opened in the coefficients at its maximum, and the polynomial can be written as $$P(z)=\sum_{n=0}^{n<b}a_n z^n+\sum_{n=b+gap}^{N+gap}a_n z^n$$ When I look at the roots of this polynomial, I find that some of them tend to concentrate around the unity circle centred at $z=(0,0)$, being the number of poles around is equal to the $gap$ variable (I consider it to be integer). In the uploaded figure I show a plot of $1/P(z)$ in the $z$ complex plane for a $gap=20$, where the roots are converted to poles and showed as bright spots. I would like to understand why the roots of the polynomial equally distribute around the unity circle.
Numerically, this distribution of zeros appear to be robust with the explicit form of the $a_n$, and only dependent on the $gap$. My intuition about this problem comes in the limit $\sigma\to0$, where we can consider only two coefficients for the polynomial (let's suppose $a_k=a_{k+1}=0.5$ and $a_n=0,\;\forall n\ne k,k+1 $). Then, when opening a gap we obtain this situation $$P_{\sigma\to0}(z)=a_k z^k+a_{k+1}z^{k+gap}$$ which has zeros in $z=0,\sqrt[gap]{1}$, consistent with the observed results. What surprises me about the problem is the robustness of the result: I recover the same behavior when increasing $\sigma$ or when adding some asymmetry (if the gap is not open at the maximum of the coefficient's distribution).
Naively, I would say that it seems to be a property of what people call topology. However, I am not an expert on the field, and I was unable to find any explanation. So, any hint of idea will be welcomed. Thank you in advance!