The limit shape of a probability measure on a tensor product of modules of the $B_n$ algebra

We study a probability measure on the integral dominant weights in the decomposition of the $N$th tensor power of the spinor representation of the Lie algebra $\mathrm{so}(2n+1)$. The probability of a dominant weight $\lambda$ is defined as 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 $\mathrm{SO}(2n+1)$-invariant measure on $\mathrm{so}(2n+1)$ induced by the Killing form. Thus, we generalize Kerov's theorem for $\mathrm{su}(n)$ to $\mathrm{so}(2n+1)$.