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I am interested in matrix integrals, and I have seen many mentions to a certain Riemann-Hilbert approach that indicate that this is a very powerful tool to can be used in this area, when coupled with the theory of orthogonal polynomials. I would like to understand it, but the sources I found were hard to follow.

It would be helpful if I could see it in action in a simple case, so I suggest what follows. Consider the matrix integral $$ f(a,N)=\int_{0}^\infty e^{-a\sum_{m=1}^\infty \frac{1}{m}{\rm Tr}(X^m)}|\Delta(X)|^2dX,$$ where $a>0$, $\Delta(X)$ is the Vandermonde and $X$ is of dimension $N$. From what I have seen, this kind of integral is amenable to the R-H approach (with a non-polynomial potential).

On the other hand, the series in the exponent is only finite if $0< X< 1$, in which case it gives $-{\rm Tr}\log(1-X)$ so using that $e^{-\infty}=0$ (I am not sure how rigorous this can be made): $$f(a,N)=\int_{0}^1 \det(1-X)^a|\Delta(X)|^2dX,$$ which is a particular case of the Selberg integral and there is an explicit solution to it, which is $$\prod_{j=0}^{N-1}\frac{\Gamma(a+j+1)j!(j+1)!}{\Gamma(a+N+j+1)}.$$

My question is: how can the R-H approach be applied to the first integral in order to produce the Selberg result?

I am interested in matrix integrals, and I have seen many mentions to a certain Riemann-Hilbert approach that indicate that this is a very powerful tool to can be used in this area, when coupled with the theory of orthogonal polynomials. I would like to understand it, but the sources I found were hard to follow.

It would be helpful if I could see it in action in a simple case, so I suggest what follows. Consider the matrix integral $$ f(a,N)=\int_{0}^\infty e^{-a\sum_{m=1}^\infty \frac{1}{m}{\rm Tr}(X^m)}|\Delta(X)|^2dX,$$ where $a>0$, $\Delta(X)$ is the Vandermonde and $X$ is diagonal of dimension $N$. I would like to know if this kind of integral is amenable to the R-H approach (with a non-polynomial potential).

The series in the exponent is only finite if $0< X< 1$, in which case it gives $-{\rm Tr}\log(1-X)$ so using that $e^{-\infty}=0$ (I am not sure how rigorous this can be made): $$f(a,N)=\int_{0}^1 \det(1-X)^a|\Delta(X)|^2dX,$$ which is a particular case of the Selberg integral and there is an explicit solution to it, which is $$\prod_{j=0}^{N-1}\frac{\Gamma(a+j+1)j!(j+1)!}{\Gamma(a+N+j+1)}.$$

My question is: how can the R-H approach be applied to the first integral in order to produce the Selberg result?

I am interested in matrix integrals, and I have seen many mentions to a certain Riemann-Hilbert approach that indicate that this is a very powerful tool to can be used in this area, when coupled with the theory of orthogonal polynomials. I would like to understand it, but the sources I found were hard to follow.

It would be helpful if I could see it in action in a simple case, so I suggest what follows. Consider the matrix integral $$ f(a,N)=\int_{0}^\infty e^{-a\sum_{m=1}^\infty \frac{1}{m}{\rm Tr}(X^m)}|\Delta(X)|^2dX,$$ where $\Delta(X)$ is the Vandermonde and $X$ is of dimension $N$. From what I have seen, this kind of integral is amenable to the R-H approach.

On the other hand, the series in the exponent is only finite if $0< X< 1$, in which case it gives $-\log(1-X)$ so using that $e^{-\infty}=0$ (I am not sure how rigorous this can be made): $$f(a,N)=\int_{0}^1 \det(1-X)^a|\Delta(X)|^2dX,$$ which is a particular case of the Serlberg integral and there is an explicit solution to it, which is $$\prod_{j=0}^{N-1}\frac{\Gamma(a+j+1)j!(j+1)!}{\Gamma(a+N+j+1)}.$$

My question is: how can the R-H approach be applied to the first integral in order to produce the Selberg result?

I am interested in matrix integrals, and I have seen many mentions to a certain Riemann-Hilbert approach that indicate that this is a very powerful tool to can be used in this area, when coupled with the theory of orthogonal polynomials. I would like to understand it, but the sources I found were hard to follow.

It would be helpful if I could see it in action in a simple case, so I suggest what follows. Consider the matrix integral $$ f(a,N)=\int_{0}^\infty e^{-a\sum_{m=1}^\infty \frac{1}{m}{\rm Tr}(X^m)}|\Delta(X)|^2dX,$$ where $a>0$, $\Delta(X)$ is the Vandermonde and $X$ is of dimension $N$. From what I have seen, this kind of integral is amenable to the R-H approach (with a non-polynomial potential).

On the other hand, the series in the exponent is only finite if $0< X< 1$, in which case it gives $-{\rm Tr}\log(1-X)$ so using that $e^{-\infty}=0$ (I am not sure how rigorous this can be made): $$f(a,N)=\int_{0}^1 \det(1-X)^a|\Delta(X)|^2dX,$$ which is a particular case of the Selberg integral and there is an explicit solution to it, which is $$\prod_{j=0}^{N-1}\frac{\Gamma(a+j+1)j!(j+1)!}{\Gamma(a+N+j+1)}.$$

My question is: how can the R-H approach be applied to the first integral in order to produce the Selberg result?