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Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where

$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$

I would like to know the following expected value

$$\lim_{n \rightarrow \infty} \mathbb E(| \det (A) |)$$

i.e., the asymptotic behavior as $n$ becomes large.

###What I tried so far

It feels like this has been done already, but after searching for quite a while I performed simulations and it looks like

$$\mathbb E(|\det(A)|) \propto e^{n f(p) }$$

where $f$ is some function of the probability $p$.

$p$." />

I would be very happy if someone knows the result or a good reference where I could look it up.

Edit: I changed the figure and added plots of the solutions

$$ \text{log} \mathbb E(|\det(A)|) \propto \frac{n}{2} \text{log} \frac{n p (1-p)}{e}$$ given by Richard Stanley and RaphaelB4 which coincide up to multiplicative terms if the factorial in Richard Stanleys solution is replaced by stirlings formula.

Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where

$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$

I would like to know the following expected value

$$\lim_{n \rightarrow \infty} \mathbb E(| \det (A) |)$$

i.e., the asymptotic behavior as $n$ becomes large.

What I tried so far

It feels like this has been done already, but after searching for quite a while I performed simulations and it looks like

$$\mathbb E(|\det(A)|) \propto e^{n f(p) }$$

where $f$ is some function of the probability $p$.

$p$." />

I would be very happy if someone knows the result or a good reference where I could look it up.

Edit: I changed the figure and added plots of the solutions

$$ \text{log} \mathbb E(|\det(A)|) \propto \frac{n}{2} \text{log} \frac{n p (1-p)}{e}$$ given by Richard Stanley and RaphaelB4 which coincide up to multiplicative terms if the factorial in Richard Stanleys solution is replaced by stirlings formula.

Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where

$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$

I would like to know the following expected value

$$\lim_{n \rightarrow \infty} \mathbb E(| \det (A) |)$$

i.e., the asymptotic behavior as $n$ becomes large.

###What I tried so far

It feels like this has been done already, but after searching for quite a while I performed simulations and it looks like

$$\mathbb E(|\det(A)|) \propto e^{n f(p) }$$

where $f$ is some function of the probability $p$.

$p$." />

I would be very happy if someone knows the result or a good reference where I could look it up.

Edit: I changed the figure and added plots of the solutions given by Richard Stanley and RaphaelB4 which coincide up to multiplicative terms if the factorial in Richard Stanleys solution is replaced by stirlings formula.

Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where

$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$

I would like to know the following expected value

$$\lim_{n \rightarrow \infty} \mathbb E(| \det (A) |)$$

i.e., the asymptotic behavior as $n$ becomes large.

###What I tried so far

It feels like this has been done already, but after searching for quite a while I performed simulations and it looks like

$$\mathbb E(|\det(A)|) \propto e^{n f(p) }$$

where $f$ is some function of the probability $p$.

$p$." />

I would be very happy if someone knows the result or a good reference where I could look it up.

Edit: I changed the figure and added plots of the solutions

$$ \text{log} \mathbb E(|\det(A)|) \propto \frac{n}{2} \text{log} \frac{n p (1-p)}{e}$$ given by Richard Stanley and RaphaelB4 which coincide up to multiplicative terms if the factorial in Richard Stanleys solution is replaced by stirlings formula.

Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where

$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$

I would like to know the following expected value

$$\lim_{n \rightarrow \infty} \mathbb E(| \det (A) |)$$

i.e., the asymptotic behavior as $n$ becomes large.

###What I tried so far

It feels like this has been done already, but after searching for quite a while I performed simulations and it looks like

$$\mathbb E(|\det(A)|) \propto e^{n f(p) }$$

where $f$ is some function of the probability $p$.

$E(|det A|)$ over $n$ for $p=0.5$ were the average is taken over 100 realizations" />

I would be very happy if someone knows the result or a good reference where I could look it up.

Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where

$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$

I would like to know the following expected value

$$\lim_{n \rightarrow \infty} \mathbb E(| \det (A) |)$$

i.e., the asymptotic behavior as $n$ becomes large.

###What I tried so far

It feels like this has been done already, but after searching for quite a while I performed simulations and it looks like

$$\mathbb E(|\det(A)|) \propto e^{n f(p) }$$

where $f$ is some function of the probability $p$.

$p$." />

I would be very happy if someone knows the result or a good reference where I could look it up.

Edit: I changed the figure and added plots of the solutions given by Richard Stanley and RaphaelB4 which coincide up to multiplicative terms if the factorial in Richard Stanleys solution is replaced by stirlings formula.