I need help explaining this analytically. If you would also check the MATLAB code for any issues. It runs well thuogh. Look at attached imag for figure that I need to explain analytically. Originally, this code generates $N$ random points in $2$D box of length $L$ using uniform distribution. Value $rc$ is the

radial distance between points, and this code ensures that the distance between values is greater than or equal to rc$.$ Now, for $L=10$ I have varied the value of rc from $0\mathrm{.}2$ to $1$ and plotted the graph shown here. This

graph is of $N($the number of points generated$)$ versus $rc.$

explain the behavior of this graph analytically. Code and images in attacheck picture. Code is also pasted below:

Intuitively, you know that $N$ will decrease as rc increases. I know that this is exponential decay, but how can I further explain this analytically? Code to copy to MATLAB:

L $=10$;

rc$\_$values $=0\mathrm{.}2$:$0\mathrm{.}1$:$1\mathrm{.}0$; $\%$ Vary rc from $0\mathrm{.}2$ to $1\mathrm{.}0$ in steps of $0\mathrm{.}1$

$\%$ Initialize an array to store the number of points for each rc value

N$\_$values $=$ zeros$($size$($rc$\_$values$))$;

for k $=1$:length$($rc$\_$values$)$

r$\_$c $=$ rc$\_$values$($k$)$;

$\%$ Generate uniformly distributed random points in $2$D

x $=$ rand$(10000,1)*$ L;

y $=$ rand$(10000,1)*$ L;

$\%$ Check the distance criteria with existing points

validPoint $=$ true$($size$($x$))$;

for i $=2$:length$($x$)$

distances $=$ sqrt$(($x$(1$:i$-1)-$ x$($i$)).^2+($y$(1$:i$-1)-$ y$($i$)).^2)$;

validPoint$($i$)=$ all$($distances $>=$ r$\_$c$)$;

end

$\%$ Count the number of valid points for the current rc value

N$\_$values$($k$)=$ sum$($validPoint$)$;

end

$\%$ Plot the results

plot$($rc$\_$values, N$\_$values, $\text{'}-$o$\text{'})$;

xlabel$(\text{'}$r$\_$c$\text{'})$;

ylabel$(\text{'}$Number of Points $($N$)\text{'})$;

title$(\text{'}$Number of Points vs$.$ r$\_$c for L$=10$ in $2$D$\text{'})$;

grid on;