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\title{Problem 3.19}
\subtitle{Random energy model}
\author{\begin{telugu}హృదయ్ దీపక్ మల్లుభొట్ల\end{telugu}}
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	Given $N$ spins (so $M = 2^N$ possible states), assign a random energy to each.
	Given $j \in \left{1, 2, \ldots, 2^N \right}$, assume each energy $E_j$ is selected with probability $P(E) = \frac{1}{\sqrt{\pi N}} e^{\flatfrac{-E^2}{N}}$.
	Gaussian with mean $0$ and standard deviation $\sqrt{\flatfrac{N}{2}}$

	\subsubsection*{(a) Microcanonical ensemble}
	Consider the states in a small range $E < E_j < E+\deltaE$.
	Let the number of such states in this range be $\Omega(E) \delta E$.
	
	Calculate the average
	\begin{equation}
		\left< \Omega(N\epsilon) \right>_{REM}
	\end{equation}
	over the ensemble of REM systems, in terms of energy per particle $\epsilon$.
	For energies per particle near zero, show that this average density of states grows exponentially as the system size $N$ grows.
	In contrast, show that $\left< \Omega(N\epsilon) \right>_{REM}$ decreases exponentially for $\epsilon = \flatfrac{E}{N} < -\epsilon_\ast$ and for $\epsilon > \epsilon_\ast$, where the limiting energy per particle
	\begin{equation}
		\epsilon_\ast = \sqrt{\log 2}.
	\end{equation}
	(Hint: as $N$ grows, the probability density $P(N \epsilon)$ decreases exponentially, which the total number of states $2^N$ grows exponentially.
	Which one wins?)

	\section{Solution} \label{sec:solution}

	
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