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Self-intersection local time $\beta_t$ is a measure of how often

a Brownian motion (or other process) crosses itself. Since Brownian

motion in the plane intersects itself so often, a renormalization

is needed in order to get something finite. LeGall proved that

$E e^{\gamma \beta_1}$ is finite for small $\gamma$ and infinite

for large $\gamma$. It turns out that the critical value is related

to the best constant in a Gagliardo-Nirenberg inequality. I will discuss

this result (joint work with Xia Chen) as well as large deviations

for $\beta_1$ and $-\beta_1$ and LILs for $\beta_t$ and $-\beta_t$.

The range of random walks is closely related to self-intersection

local times, and I will also discuss joint work with Jay Rosen

making this idea precise.