The columns are distributed as $X^N/||X^N||$, where $X^N = (X_1^N,\dots,X^N_N)$ is the standard Gaussian vector in $\mathbb{R}^N$. The product of two different columns, $$ R_N:=\frac{\sum_{i=1}^N X_i^NY_i^N}{\left(\sum_{i=1}^N (X_i^N)^2\sum_{i=1}^N (X_i^N)^2\right)^{1/2}} =\frac{\frac1N\sum_{i=1}^N X_i^NY_i^N}{\left(\frac1N\sum_{i=1}^N (X_i^N)^2 \frac1N\sum_{i=1}^N (X_i^N)^2\right)^{1/2}}, $$ converges to $0$ almost surely$^*$ by SLLN. Alternatively, you can understand this product as an empirical correlation function of $X$ and $Y$.

$^*$ It is not very clear what almost surely means here, since the joint characteristics for matrices for different $N$ are not given. But I believe that this is true for any joint distribution, that is, for any $C>0$, $\sum_{N\ge 1}\mathbb{P}(|R_N|>C)<\infty$. This should be possible to extract from properties of Gaussian distribution or from concentration inequalities.

The columns are distributed as $X^N/||X^N||$, where $X^N = (X_1^N,\dots,X^N_N)$ is the standard Gaussian vector in $\mathbb{R}^N$. The product of two different columns, $$ R_N:=\frac{\sum_{i=1}^N X_i^NY_i^N}{\left(\sum_{i=1}^N (X_i^N)^2\sum_{i=1}^N (X_i^N)^2\right)^{1/2}} =\frac{\frac1N\sum_{i=1}^N X_i^NY_i^N}{\left(\frac1N\sum_{i=1}^N (X_i^N)^2 \frac1N\sum_{i=1}^N (X_i^N)^2\right)^{1/2}}, $$ converges to $0$ almost surely$^*$ by SLLN. Alternatively, you can understand this product as an empirical correlation function of the two samples: $X$ and $Y$.

The distribution is generalized chi-square, if this helps.

$^*$ It is not very clear what almost surely means here, since the joint characteristics for matrices for different $N$ are not given. But I believe that this is true for any joint distribution, that is, for any $C>0$, $\sum_{N\ge 1}\mathbb{P}(|R_N|>C)<\infty$. This should be possible to extract from properties of Gaussian distribution or from concentration inequalities.