We prove a relativization of the Alon Second Eigenvalue Conjecture for all
d-regular base graphs, B, with dβ₯3: for any Ο΅>0, we show that
a random covering map of degree n to B has a new eigenvalue greater than
2dβ1β+Ο΅ in absolute value with probability O(1/n).
Furthermore, if B is a Ramanujan graph, we show that this probability is
proportional to nβΞ·fundβ(B), where Ξ·fundβ(B)
is an integer depending on B, which can be computed by a finite algorithm for
any fixed B. For any d-regular graph, B, Ξ·fundβ(B) is
greater than dβ1β.
Our proof introduces a number of ideas that simplify and strengthen the
methods of Friedman's proof of the original conjecture of Alon. The most
significant new idea is that of a ``certified trace,'' which is not only
greatly simplifies our trace methods, but is the reason we can obtain the
nβΞ·fundβ(B) estimate above. This estimate represents an
improvement over Friedman's results of the original Alon conjecture for random
d-regular graphs, for certain values of d