Mass-gap estimate for finite-dimensional sigma-model approximations

Let $S^k$ be the target sphere of radius $R$, and let $\tilde{\mathcal{L}}_{N,U}(S^k)$ be the manifold with boundary obtained by removing a tubular neighborhood $U(\mathcal{L}_{N-1}(S^k))$ from the finite-dimensional approximation $\mathcal{L}_N(S^k)$. Let $\lambda_1$ and $\lambda_2$ be the first two eigenvalues of the Neumann or Dirichlet problem for the Schrödinger operator on $\tilde{\mathcal{L}}_{N,U}(S^k)$, and set $\Gamma_N=\lambda_2-\lambda_1$. Mass-gap estimate. There are constants $C_1,C_2>0$ such that

$$\Gamma(N)\geq C_1N^2e^{-C_2R^2}.$$

Such an estimate would provide a uniform mass-gap lower bound for the finite-dimensional sigma-model approximations as $N\to\infty$. The statement leaves open which boundary condition, Neumann or Dirichlet, is intended and does not specify the dependence of the constants on the remaining parameters.