The successive algebraic-power conjectures for basic models over regular graphs

Let $B$ be a $d$-regular graph and let $\{\mathcal{C}_n(B)\}_{n\in N}$ be one of the basic models over $B$, with tangle power $\tau_{\rm tang}$ and algebraic power $\tau_{\rm alg}$. For $\epsilon>0$, ${\rm NonAlon}_d(G;\epsilon)$ denotes the number of new eigenvalues outside the Alon bound by more than $\epsilon$. Successive algebraic-power conjectures. The following claims are conjectured:

1. For sufficiently small $\epsilon>0$, there are constants $C,C'$ such that for sufficiently large $n$,

$$C'n^{-\tau_{\rm tang}}\leq \operatorname{Prob}_{G\in\mathcal{C}_n(B)}\bigl[{\rm NonAlon}_d(G;\epsilon)>0\bigr]\leq Cn^{-\tau_{\rm tang}}.$$

2. $\tau_{\rm tang}\leq \tau_{\rm alg}-1$.

3. $\tau_{\rm alg}=+\infty$.

These are presented as successively stronger conjectures related to the theorem proving infinite algebraic power for basic models over regular Ramanujan graphs; the general conjectures remain open in the supplied text.