Probability measure near the boundary of tensor power decomposition for $\mathfrak{so}_{2n+1}$

Character measure is a probability measure on irreducible representations of a semisimple Lie algebra. It appears from the decomposition into irreducibles of tensor power of a fundamental representation. In this paper we calculate the asymptotics of character measure on representations of $\mathfrak{so}_{2n+1}$ in the regime near the boundary of weight diagram. We find out that it converges to a Poisson-type distribution.