I am trying to evaluate or at least obtains bounds for the following integral for $0<\gamma^{2}<2$

$$ \int_{[0,1]^{2n}}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{2-\gamma^{2}}\prod_{1\leq j<k\leq n}|e^{i2\pi \phi_{k}}-e^{i2\pi \phi_{j}}|^{2-\gamma^{2}} \prod_{1\leq j,k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \phi_{j}}|^{-\gamma^{2}}\prod_{i}^{n}d\theta_{i}d\phi_{i} .$$

Any suggestions on how to proceed?

Attempts

1)The Dixon-Anderson identity is (cf "Log-Gases and RMT" chapter 4):

$$ \int_{R_{r}}\prod_{1\leq j<k\leq rn}|e^{i2\pi \phi_{k}}-e^{i2\pi \phi_{j}}|^{2/(r+1)} \prod_{1\leq j\leq rn,1\leq k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \phi_{j}}|^{-\alpha_{j}}\prod_{i}^{rn}d\phi_{i} =c_{r}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{r(\alpha_{j}+\alpha_{k}-2/(r+1))},$$

where $R_{r}:=\theta_{j-1}<\psi_{(r-1)j+1}<\psi_{(r-1)j+2}<...<\psi_{(r-1)j}<\theta_{j}.$

If we take $r:=\gamma^{2}/2/(1-\gamma^{2}/2)$ and $\alpha_{j}=\gamma^{2}$ we get something similar with the difference $r\cdot n$ instead of n. So by taking $\gamma^{2}>(<)1$ I am trying to get bounds in terms of the Dixon-Anderson integral. And then we can use the Selberg integral.

2)DOTSENKO AND FATEEV evaluated the following integral (cf same book section 4.5)

$$ \int_{[0,1]^{2n}}\prod_{i}^{n}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{2a}\prod_{1\leq j<k\leq n}|e^{i2\pi \phi_{k}}-e^{i2\pi \phi_{j}}|^{2b} \prod_{1\leq j,k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \phi_{j}}|^{-2}\prod_{i}^{n}d\theta_{i}d\phi_{i} $$

where $a*b=1$. This works if $\gamma=2$, which is not possible in our case.

I am trying to evaluate or at least obtains bounds for the following integral for $0<\gamma^{2}<2$

$$ \int_{[0,1]^{2n}}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{2-\gamma^{2}}\prod_{1\leq j<k\leq n}|e^{i2\pi \phi_{k}}-e^{i2\pi \phi_{j}}|^{2-\gamma^{2}} \prod_{1\leq j,k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \phi_{j}}|^{-\gamma^{2}}\prod_{i}^{n}d\theta_{i}d\phi_{i} .$$

Q: Any suggestions on how to proceed?

Attempts

1)The Dixon-Anderson identity is (cf "Log-Gases and RMT" chapter 4):

$$ \int_{R_{r}}\prod_{1\leq j<k\leq rn}|e^{i2\pi \phi_{k}}-e^{i2\pi \phi_{j}}|^{2/(r+1)} \prod_{1\leq j\leq rn,1\leq k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \phi_{j}}|^{-\alpha_{j}}\prod_{i}^{rn}d\phi_{i} =c_{r}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{r(\alpha_{j}+\alpha_{k}-2/(r+1))},$$

where $R_{r}:=\theta_{j-1}<\psi_{(r-1)j+1}<\psi_{(r-1)j+2}<...<\psi_{(r-1)j}<\theta_{j}.$

If we take $r:=\gamma^{2}/2/(1-\gamma^{2}/2)$ and $\alpha_{j}=\gamma^{2}$ we get something similar with the difference $r\cdot n$ instead of n. So by taking $\gamma^{2}>(<)1$ I am trying to get bounds in terms of the Dixon-Anderson integral. And then we can use the Selberg integral.

2)DOTSENKO AND FATEEV evaluated the following integral (cf same book section 4.5)

$$ \int_{[0,1]^{2n}}\prod_{i}^{n}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{2a}\prod_{1\leq j<k\leq n}|e^{i2\pi \phi_{k}}-e^{i2\pi \phi_{j}}|^{2b} \prod_{1\leq j,k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \phi_{j}}|^{-2}\prod_{i}^{n}d\theta_{i}d\phi_{i} $$

where $a*b=1$. This works if $\gamma=2$, which is not possible in our case.

I am trying to evaluate or at least obtains bounds for the following integral for $0<\gamma^{2}<2$

$$ \int_{[0,1]^{2n}}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{2-\gamma^{2}}\prod_{1\leq j<k\leq n}|e^{i2\pi \phi_{k}}-e^{i2\pi \phi_{j}}|^{2-\gamma^{2}} \prod_{1\leq j,k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \phi_{j}}|^{-\gamma^{2}}\prod_{i}^{n}d\theta_{i}d\phi_{i} .$$

Attempts

1)The Dixon-Anderson identity is (cf "Log-Gases and RMT" chapter 4):

$$ \int_{R_{r}}\prod_{1\leq j<k\leq rn}|e^{i2\pi \phi_{k}}-e^{i2\pi \phi_{j}}|^{2/(r+1)} \prod_{1\leq j\leq rn,1\leq k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \phi_{j}}|^{-\alpha_{j}}\prod_{i}^{rn}d\phi_{i} =c_{r}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{r(\alpha_{j}+\alpha_{k}-2/(r+1))},$$

where $R_{r}:=\theta_{j-1}<\psi_{(r-1)j+1}<\psi_{(r-1)j+2}<...<\psi_{(r-1)j}<\theta_{j}.$

If we take $r:=\gamma^{2}/2/(1-\gamma^{2}/2)$ and $\alpha_{j}=\gamma^{2}$ we get something similar with the difference $r\cdot n$ instead of n. So by taking $\gamma^{2}>(<)1$ I am trying to get bounds in terms of the Dixon-Anderson integral. And then we can use the Selberg integral.

2)DOTSENKO AND FATEEV evaluated the following integral (cf same book section 4.5)

$$ \int_{[0,1]^{2n}}\prod_{i}^{n}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{2a}\prod_{1\leq j<k\leq n}|e^{i2\pi \phi_{k}}-e^{i2\pi \phi_{j}}|^{2b} \prod_{1\leq j,k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \phi_{j}}|^{-2}\prod_{i}^{n}d\theta_{i}d\phi_{i} $$

where $a*b=1$. This works if $\gamma=2$, which is not possible in our case.

I am trying to evaluate or at least obtains bounds for the following integral for $0<\gamma^{2}<2$

$$ \int_{[0,1]^{2n}}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{2-\gamma^{2}}\prod_{1\leq j<k\leq n}|e^{i2\pi \phi_{k}}-e^{i2\pi \phi_{j}}|^{2-\gamma^{2}} \prod_{1\leq j,k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \phi_{j}}|^{-\gamma^{2}}\prod_{i}^{n}d\theta_{i}d\phi_{i} .$$

Any suggestions on how to proceed?

Attempts

1)The Dixon-Anderson identity is (cf "Log-Gases and RMT" chapter 4):

$$ \int_{R_{r}}\prod_{1\leq j<k\leq rn}|e^{i2\pi \phi_{k}}-e^{i2\pi \phi_{j}}|^{2/(r+1)} \prod_{1\leq j\leq rn,1\leq k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \phi_{j}}|^{-\alpha_{j}}\prod_{i}^{rn}d\phi_{i} =c_{r}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{r(\alpha_{j}+\alpha_{k}-2/(r+1))},$$

where $R_{r}:=\theta_{j-1}<\psi_{(r-1)j+1}<\psi_{(r-1)j+2}<...<\psi_{(r-1)j}<\theta_{j}.$

If we take $r:=\gamma^{2}/2/(1-\gamma^{2}/2)$ and $\alpha_{j}=\gamma^{2}$ we get something similar with the difference $r\cdot n$ instead of n. So by taking $\gamma^{2}>(<)1$ I am trying to get bounds in terms of the Dixon-Anderson integral. And then we can use the Selberg integral.

2)DOTSENKO AND FATEEV evaluated the following integral (cf same book section 4.5)

$$ \int_{[0,1]^{2n}}\prod_{i}^{n}\prod_{1\leq j<k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \theta_{j}}|^{2a}\prod_{1\leq j<k\leq n}|e^{i2\pi \phi_{k}}-e^{i2\pi \phi_{j}}|^{2b} \prod_{1\leq j,k\leq n}|e^{i2\pi \theta_{k}}-e^{i2\pi \phi_{j}}|^{-2}\prod_{i}^{n}d\theta_{i}d\phi_{i} $$

where $a*b=1$. This works if $\gamma=2$, which is not possible in our case.