Nondegeneracy of random regular Tanner graphs with logarithmic girth

Let $d_c>d_v\geq 3$ be integers satisfying $d_v<\beta_{d_c}d_c$, where $\beta_{d_c}$ is Calkin's threshold. For a positive constant $\lambda$, let $\Gamma_{\lambda}$ be the ensemble of $(d_v,d_c)$-regular Tanner graphs on $n$ variable nodes with girth at least $\lambda\log n$. Nondegeneracy conjecture. There is a sufficiently small constant $\lambda>0$ such that, for every constant $c>0$, a random graph $G$ from $\Gamma_{\lambda}$ is $(c\log n,\omega(1))$-nondegenerate with high probability. This conjecture concerns the rigidity condition: for graphs of logarithmic girth and minimum check degree at least three, the stated nondegeneracy property is equivalent to the simpler rigidity condition discussed in the paper.