Limit shape of probability measure on tensor product of $B_n$ algebra modules

We study a probability measure on integral dominant weights in the decomposition of $N$-th tensor power of spinor representation of the Lie algebra $so(2n+1)$. The probability of the dominant weight $\lambda$ is defined as the ratio of the dimension of the irreducible component of $\lambda$ divided by the total dimension $2^{nN}$ of the tensor power. We prove that as $N\to \infty$ the measure weakly converges to the radial part of the $SO(2n+1)$-invariant measure on $so(2n+1)$ induced by the Killing form. Thus, we generalize Kerov's theorem for $su(n)$ to $so(2n+1)$.