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Write $V(a)$ for the determinant $\prod_{0\leq i<j\leq n-1} |a_i-a_j|$. Selberg's formula tells you that

$$\int_0^1 \cdots \int_0^1 V(a)^{2\beta} \prod_{i=0}^{n-1} da_i= n! \prod_{j=0}^{n-1} \frac{(\Gamma(1+j\beta))^2 \cdot \Gamma((j+1)\beta)} {\Gamma(2+(n+j-1)\beta)\cdot \Gamma(\beta)}=:A(n,\beta)$$

Thus the asymptotics you seek are given by $\lim_{\beta\to\infty} A(n,\beta)^{1/2\beta}$, which can be read from known asymptotics for the Gamma function. I did not try to perform the actual computation.

Remark: The constant $-2b^2$ is the maximum of the logarithmic energy $$\int \log |x-y| \mu(dx) \mu(dy) $$ over all probability measures supported on $[0,1]$. I am sure that this maximizer has been computed somewhere; Maybe it appears in Saff and Totitc's book, which I do not have access to at the moment.

Write $V(a)$ for the determinant $\prod_{0\leq i<j\leq n-1} |a_i-a_j|$. Selberg's formula tells you that

$$\int_0^1 \cdots \int_0^1 V(a)^{2\beta} \prod_{i=0}^{n-1} da_i= n! \prod_{j=0}^{n-1} \frac{(\Gamma(1+j\beta))^2 \cdot \Gamma((j+1)\beta)} {\Gamma(2+(n+j-1)\beta)\cdot \Gamma(\beta)}=:A(n,\beta)$$

Thus the asymptotics you seek are given by $\lim_{\beta\to\infty} A(n,\beta)^{1/2\beta}$, which can be read from known asymptotics for the Gamma function. I did not try to perform the actual computation.

Remark: The constant $-2b^2$ is the maximum of the logarithmic energy $$\int \log |x-y| \mu(dx) \mu(dy) $$ over all probability measures supported on $[0,1]$. I am sure that this maximizer has been computed somewhere; Maybe it appears in Saff and Totik's book, which I do not have access to at the moment.

Write $V(a)$ for the determinant $\prod_{0\leq i<j\leq n-1} |a_i-a_j|$. Selberg's formula tells you that

$$\int_0^1 \cdots \int_0^1 V(a)^{2\beta} \prod_{i=0}^{n-1} da_i= n! \prod_{j=0}^{n-1} \frac{(\Gamma(1+j\beta))^2 \cdot \Gamma((j+1)\beta)} {\Gamma(2+(n+j-1)\beta)\cdot \Gamma(\beta)}=:A(n,\beta)$$

Thus the asymptotics you seek are given by $\lim_{\beta\to\infty} A(n,\beta)^{1/2\beta}$, which can be read from known asymptotics for the Gamma function. I did not try to perform the actual computation.

Remark: The constant $2b^2$ is the maximum of the logarithmic energy $$\int \log |x-y| \mu(dx) \mu(dy) $$ over all probability measures supported on $[0,1]$. I am sure that this maximizer has been computed somewhere; Maybe it appears in Saff and Totitc's book, which I do not have access to at the moment.

Write $V(a)$ for the determinant $\prod_{0\leq i<j\leq n-1} |a_i-a_j|$. Selberg's formula tells you that

$$\int_0^1 \cdots \int_0^1 V(a)^{2\beta} \prod_{i=0}^{n-1} da_i= n! \prod_{j=0}^{n-1} \frac{(\Gamma(1+j\beta))^2 \cdot \Gamma((j+1)\beta)} {\Gamma(2+(n+j-1)\beta)\cdot \Gamma(\beta)}=:A(n,\beta)$$

Thus the asymptotics you seek are given by $\lim_{\beta\to\infty} A(n,\beta)^{1/2\beta}$, which can be read from known asymptotics for the Gamma function. I did not try to perform the actual computation.

Remark: The constant $-2b^2$ is the maximum of the logarithmic energy $$\int \log |x-y| \mu(dx) \mu(dy) $$ over all probability measures supported on $[0,1]$. I am sure that this maximizer has been computed somewhere; Maybe it appears in Saff and Totitc's book, which I do not have access to at the moment.

Write $V(a)$ for the determinant $\prod_{0\leq i<j\leq n-1} |a_i-a_j|$. Selberg's formula tells you that

$$\int_0^1 \cdots \int_0^1 V(a)^{2\beta} \prod_{i=0}^{n-1} da_i= n! \prod_{j=0}^{n-1} \frac{(\Gamma(1+j\beta))^2 \cdot \Gamma((j+1)\beta)} {\Gamma(2+(n+j-1)\beta)\cdot \Gamma(\beta)}=:A(n,\beta)$$

Thus the asymptotics you seek are given by $\lim_{\beta\to\infty} A(n,\beta)^{1/2\beta}$, which can be read from known asymptotics for the Gamma function. I did not try to perform the actual computation.

Remark: The constant $b$ is the maximum of the logarithmic energy $$\int \log |x-y| \mu(dx) \mu(dy) $$ over all probability measures supported on $[0,1]$. I am sure that this maximizer has been computed somewhere; Maybe it appears in Saff and Totitc's book, which I do not have access to at the moment.

Write $V(a)$ for the determinant $\prod_{0\leq i<j\leq n-1} |a_i-a_j|$. Selberg's formula tells you that

$$\int_0^1 \cdots \int_0^1 V(a)^{2\beta} \prod_{i=0}^{n-1} da_i= n! \prod_{j=0}^{n-1} \frac{(\Gamma(1+j\beta))^2 \cdot \Gamma((j+1)\beta)} {\Gamma(2+(n+j-1)\beta)\cdot \Gamma(\beta)}=:A(n,\beta)$$

Thus the asymptotics you seek are given by $\lim_{\beta\to\infty} A(n,\beta)^{1/2\beta}$, which can be read from known asymptotics for the Gamma function. I did not try to perform the actual computation.

Remark: The constant $2b^2$ is the maximum of the logarithmic energy $$\int \log |x-y| \mu(dx) \mu(dy) $$ over all probability measures supported on $[0,1]$. I am sure that this maximizer has been computed somewhere; Maybe it appears in Saff and Totitc's book, which I do not have access to at the moment.