TITLE: Primes isolated by large gaps to either side
QUESTION [7 upvotes]: Say that the $n$-th prime $p_n$ is isolated to degree $k$ 
(my notation) if
the prime gap to either side is larger than $\log p_n$ to the $k$-th power:
\begin{eqnarray*}
p_n - p_{n-1} & > & (\log p_n)^k \;,\\
p_{n+1} - p_n & > & (\log p_n)^k \;,
\end{eqnarray*}
where $\log$ is the natural log.
Examples. 

For $k=1.5$, 
$p_{4059}=38501$ is isolated because
$$(p_{n-1},p_n,p_{n+1}) = (38461,38501,38543)\;,$$
and the gaps of $40$ and $42$ both exceed 
$(\log 38501)^{3/2} \approx 10.6^{1.5} \approx 34.3$.
For $k=1.6$,
$p_{722697}=10938023$ is isolated because
$$(p_{n-1},p_n,p_{n+1}) = (10937921,10938023,10938119)\;,$$
and the gaps of $102$ and $96$ both exceed $(\log 10938023)^{1.6} \approx 86.2$.
For $k=1.7$, I find no isolated primes in the first
$10$-million primes.
(The $10$-th million prime is $179424673$.)
Among the first
$10$-million primes, about $13$% are isolated to degree $k = 1$,
and $73$% are isolated to degree $k=\frac{1}{2}$.


Q. For which $k$ are there an infinite number of isolated primes of degree $k$?

REPLY [4 votes]: Currently it is not even known, for any $k>1$, if $p_{n+1}-p_n>(\log p_n)^k$ holds infinitely often. The best known result in this direction is due to Ford, Green, Konyagin, Maynard, Tao, see here.
On the other hand, as Jeremy Rouse explained in two comments, it is expected that the original inequalities hold infinitely often for any $k<2$.