Let $(X_n)_n$ be a simple random walk in $Z^2$ starting from the origin which is killed upon leaving the ball of radius n centred at the origin, let $R_n$ be the range of the walk, let $ L_x $ be the number of visits at $x$.
Is it true that for every $K$ there exists $\delta, c_1,c_2$ positive such that, $$ P( \, | \{ x \in R_n \, : \, L_x > K \} | < \delta n^2 ) \leq c_1 e^{ - c_2 n}? $$
The question is whether it is unlikely that the fraction of the vertices of the range with sufficiently large local time is too small. Note that the walk takes time O(n^2) to leave the ball. I would expect the claim to be true since it is known that $ E(L_x) = O( \log(n) / |x|_2) $ for for every vertex x such that $|x|_2\leq n$. Namely, in expectation we expect a diverging number of visits inside the ball, so I think it should be unlikely to visit many vertices just few times. But how can this probability be quantified?