The proposed small-gamma centered-adjacency tail has the opposite limit

Throughout the facts and proof, set $k=\lceil L\mathrm{/}1000\rceil$, the least integer at least $L\mathrm{/}1000$, as in the logarithmic parameter lemma (Explicit parameters for the small-gamma logarithmic density).

A vertex of degree at least $k$ forces $\mathrm{\parallel }A-pJ\mathrm{\parallel }\ge (1-p)\sqrt{k}$ (A centered adjacency column retains a high-degree lower bound).

For this same $k$, the parameters satisfy $0<p<1\mathrm{/}2$ and $L\mathrm{/}1000\le k\le n\mathrm{/}2$ (Explicit parameters for the small-gamma logarithmic density).

A vertex of degree at least $k$ occurs with probability at least $1\mathrm{/}[1+2\exp (-49L\mathrm{/}50)]$, tending to one (Independent cross-degrees produce a large degree at logarithmic density).

For nonnegative $x$, $\exp (x)$ is bounded below by every finite partial sum of its positive series (The real exponential function and the number e by a power series).

The exponential is positive and $\exp (-x)=1\mathrm{/}\exp (x)$ (The exponential is positive and satisfies exp(-x)=1/exp(x)).

The exponential is strictly increasing (The exponential function is strictly increasing).

For positive $n$, the real power $n^{-\gamma }$ equals $\exp (-\gamma \log n)$ (Real powers for positive bases, with the zero-base positive-exponent convention).

The law $G(n,p)$ is the finite simple graph on $[n]$ with mutually independent Bernoulli$(p)$ edge indicators, one for each two-element vertex subset (The Erdős-Rényi finite random graph G(n,p)).

For $n>0$, $L=\log n$ means $\exp (L)=n$ (The natural logarithm as the inverse of the exponential function).

For a matrix acting on the Euclidean inner product space ${\mathbb{R}}^n$, $\mathrm{\parallel }B\mathrm{\parallel }={\max }_{\mathrm{\parallel }v{\mathrm{\parallel }}_2=1}\mathrm{\parallel }Bv{\mathrm{\parallel }}_2$ (The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖).

An event is a subset of the finite outcome space and its probability is the sum of its nonnegative outcome weights, whose total is one. Consequently $0\le \mathbb{P}(B)\le 1$, and $\mathbb{P}(B)\le \mathbb{P}(C)$ whenever $B\subseteq C$ (Finite probability spaces, outcome weights, events, and event probabilities).

Two distinct vertices are adjacent when they form an edge; the degree of a vertex is the cardinality of its open neighborhood, the set of its adjacent vertices (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).