Multiple intersection exponents

Let $p\ge 2$, $n_1\le \mathrm{.}\mathrm{.}\mathrm{.}\le n_p$ be positive integers and $B_1^1, …, B_{n_1}^1; …; B_1^p, …, B_{n_p}^{p}$ be independent planar Brownian motions started uniformly on the boundary of the unit circle. We define a $p$-fold intersection exponent $\varsigma (n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_p)$, as the exponential rate of decay of the probability that the packets ${\bigcup }_{j=1}^{n_i}{B}_j^i[0,t^2]$, $i=1,\mathrm{.}\mathrm{.}\mathrm{.},p$, have no joint intersection. The case $p=2$ is well-known and, following two decades of numerical and mathematical activity, Lawler, Schramm and Werner (2001) rigorously identified precise values for these exponents. The exponents have not been investigated so far for $p>2$. We present an extensive mathematical and numerical study, leading to an exact formula in the case $n_1=1$, $n_2=2$, and several interesting conjectures for other cases.