The Ben Arous–Guionnet Large Deviation Principle for the Empirical Spectral Measure

Intuition Beginner

You already know that the eigenvalues of a large Gaussian random matrix arrange themselves into a smooth, predictable shape — the semicircle. Resample the matrix and the histogram of eigenvalues barely twitches: the shape is locked in. This unit asks the sharper question. The histogram almost never strays from the semicircle, but how unlikely is it to stray? If you demanded that the eigenvalues pile up into some completely different shape — say a flat slab, or two separated lumps — what would you have to pay?

The answer is an exponential price, and the size of that price is the whole story. Think of the eigenvalues as a gas of charged beads on a wire. Like charges, they repel; a confining field pulls them back toward the center. Left alone, the gas settles into one balanced shape that minimizes its total energy, and that shape is the semicircle. To force the gas into any other shape, you fight its own energetics, and the chance of seeing that wrong shape shrinks like a falling exponential whose exponent is the extra energy the wrong shape carries.

What is new and strong here is the speed of the decay. For a single average of independent samples, the unlikeliness grows with the number of samples. For the whole eigenvalue cloud it grows with the number of samples squared, because every pair of eigenvalues interacts with every other. So the semicircle is not merely typical — it is overwhelmingly, doubly-exponentially enforced, and the energy of a shape measures exactly how hard the gas resists being pushed there.

Visual Beginner

Picture a bowl-shaped energy landscape sitting over the space of possible eigenvalue shapes. The very bottom of the bowl is the semicircle: the one shape the charged gas relaxes into on its own. Every other shape sits higher up the wall of the bowl, and its height above the bottom is the extra energy you pay to force the gas there. The chance of seeing a given shape falls off like an exponential of minus that height, multiplied by the matrix size squared.

   energy landscape over "shapes of the eigenvalue cloud"
   energy
     ^            wrong shape B
     |           (two lumps)
     |   wrong      .o.            wrong shape C
     |  shape A    /   \          (flat slab)
     |   .o.      /     \          .o.
     |  /   \    /       \        /   \
     | /     \__/         \______/     \
     |/                                 \
     +----*-------------------------------*----> shapes
          \____   semicircle = bowl floor
               (lowest energy, the typical shape)

   chance of shape  ~  exp( - n^2 * height above the floor )
   height of shape  =  its energy  -  energy of the semicircle

The bowl has a single lowest point because the energy combines a repulsion that punishes crowding with a confinement that punishes spreading, and those two pulls balance at exactly one shape. That balance point is the semicircle, and the height of the wall above it is the rate at which every other shape is suppressed.

Worked example Beginner

We compare the energy of two simple eigenvalue shapes for a confined charged gas and see which one the gas prefers, reproducing in miniature why a balanced shape wins.

Step 1. Set up two beads. Put two eigenvalues at positions $a$ and $b$. The energy has two parts: a confinement part that adds up the squares of the positions, $\frac{1}{4}(a^2+b^2)$, and a repulsion part that subtracts the logarithm of the gap, $-\log |a-b|$. Repulsion lowers the energy when the beads are far apart; confinement raises it when they wander out.

Step 2. A spread-out shape. Place the beads at $a=-2$ and $b=+2$. Confinement gives $\frac{1}{4}(4+4)=2$. The gap is $4$, so repulsion gives $-\log 4=-1.386$. Total energy: $2-1.386=0.614$.

Step 3. A balanced shape. Now place them closer, at $a=-1$ and $b=+1$. Confinement gives $\frac{1}{4}(1+1)=0.5$. The gap is $2$, so repulsion gives $-\log 2=-0.693$. Total energy: $0.5-0.693=-0.193$.

Step 4. A crowded shape. Push them together at $a=-0.3$ and $b=+0.3$. Confinement gives $\frac{1}{4}(0.09+0.09)=0.045$. The gap is $0.6$, so repulsion gives $-\log 0.6=0.511$ — now positive, because a small gap is expensive. Total energy: $0.045+0.511=0.556$.

Step 5. What this tells us. The balanced shape at $\pm 1$ has the lowest energy of the three: spreading too far costs confinement, crowding too close costs repulsion, and the middle wins. The true equilibrium shape — the semicircle for a full matrix — is the one balancing these costs across all the eigenvalues at once, and any shape with higher energy is exponentially less likely to appear.

Check your understanding Beginner

Formal definition Intermediate+

Fix $\beta \in \{1,2,4\}$ (or any $\beta >0$ for the log-gas) and let the eigenvalues ${\lambda }_1,\dots ,{\lambda }_n$ be distributed according to the Gaussian $\beta$-ensemble joint density 37.08.03 $$ p_{n,\beta}(\lambda) = \frac{1}{Z_{n,\beta}},\prod_{i<j}|\lambda_i - \lambda_j|^{\beta},\exp!\Big(-\tfrac{\beta n}{4}\sum_{i=1}^n \lambda_i^2\Big). $$ The empirical spectral measure is the random probability measure $$ L_n ;:=; \frac{1}{n}\sum_{i=1}^n \delta_{\lambda_i} ;\in; \mathcal{M}_1(\mathbb{R}), $$ where ${\mathcal{M}}_1(\mathbb{R})$ is the space of Borel probability measures on $\mathbb{R}$ equipped with the weak topology (the coarsest making $\mu \mapsto \int \varphi d\mu$ continuous for every bounded continuous $\varphi$), under which ${\mathcal{M}}_1(\mathbb{R})$ is Polish. This is the same measure-valued statistic of 37.07.05, now built from dependent — indeed strongly repelling — variables.

The rate functional is the logarithmic-energy / non-commutative entropy functional. For a confining potential $V(x)=\frac{1}{2}{x}^2$, define on ${\mathcal{M}}_1(\mathbb{R})$ $$ \mathcal{E}(\mu) ;:=; \int_{\mathbb{R}} \frac{x^2}{2},d\mu(x) ;-; \iint_{\mathbb{R}^2} \log|x - y|,d\mu(x),d\mu(y), $$ the weighted logarithmic energy of $\mu$ in the external field $V$. It splits into a confinement term (the $V$-integral) and the logarithmic energy $\Sigma (\mu ):=\iint \log |x-y|d\mu (x)d\mu (y)$, Voiculescu's free-entropy quantity ^{[Voiculescu 1993]}. The rate functional of the LDP is the normalised energy $$ I_\beta(\mu) ;:=; \frac{\beta}{2}\Big(\mathcal{E}(\mu) - \inf_{\nu \in \mathcal{M}1(\mathbb{R})}\mathcal{E}(\nu)\Big), $$ shifted so that $\inf\mu I_\beta = 0$.Theshiftconstant$c_\beta := \frac{\beta}{2}\inf_\nu \mathcal{E}(\nu)$isfinite,and$\mathcal{E}$iswelldefinedin$(-\infty, +\infty]$onceonechecksthedoubleintegraldoesnotproducea$+\infty - \infty$indeterminacy(thenegativepartof$\log|x-y|$ is controlled by the confinement; see the proof set).

Definition (the Ben Arous–Guionnet LDP). The laws of $\{L_n\}$ satisfy a large deviation principle 37.07.01 on ${\mathcal{M}}_1(\mathbb{R})$ in the weak topology at speed $a_n=1/n^2$ with good rate function $I_{\beta }$: for every Borel $\Gamma \subseteq {\mathcal{M}}_1(\mathbb{R})$, $$ -\inf_{\mu\in\Gamma^\circ} I_\beta(\mu) ;\leq; \liminf_{n}\tfrac{1}{n^2}\log\mathbb{P}(L_n\in\Gamma) ;\leq; \limsup_{n}\tfrac{1}{n^2}\log\mathbb{P}(L_n\in\Gamma) ;\leq; -\inf_{\mu\in\overline{\Gamma}} I_\beta(\mu). $$ A good rate function is non-negative, lower-semicontinuous, and has compact sublevel sets $\{I_{\beta }\le \alpha \}$; all three hold for $I_{\beta }$ in the weak topology.

The unique minimiser is the semicircle law $\sigma (dx)=\frac{1}{2\pi }\sqrt{4-x^2}{1}_{[-2,2]}(x)dx$, so $I_{\beta }(\mu )=0$ if and only if $\mu =\sigma$, recovering the law of large numbers $L_n\to \sigma$ from the LDP as the statement that the rate vanishes only there.

Counterexamples to common slips

• The speed is $n^2$, not $n$. Sanov 37.07.05 prices the empirical measure of independent samples at speed $n$ with relative-entropy rate. Here the eigenvalues interact through $|\Delta {|}^{\beta }=\exp (\beta {\sum }_{i<j}\log |{\lambda }_i-{\lambda }_j|)$, an energy of order $n^2$, so the correct normalisation is $1/n^2$ and the rate is a quadratic energy, not a linear entropy. Using speed $n$ gives a degenerate (identically $+\infty$ or $0$) statement.
• $I_{\beta }$ is not the relative entropy $H(\mu \Vert \sigma )$. The rate is the logarithmic energy, a quadratic form in $\mu$, fundamentally different from the linear-in-$\mu$ relative entropy of Sanov. The two agree only in spirit (both vanish at the equilibrium law); their analytic forms are unrelated.
• The minimiser depends on the potential, not on $\beta$. The factor $\frac{\beta }{2}$ in $I_{\beta }$ scales the whole rate but does not move its minimiser: the equilibrium measure solves a $\beta$-free variational problem and equals the semicircle for the quadratic $V$. Changing $V$ changes the minimiser; changing $\beta$ only rescales how strongly deviations are penalised.
• Concentration is not the LDP. The log-Sobolev / Herbst concentration of 37.08.07 gives a sub-Gaussian bound $\mathbb{P}(\text{deviation}\ge t)\le 2e^{-c{n}^2{t}^2}$ on linear statistics — the right speed but a Gaussian envelope, not the exact exponential rate. The LDP supplies the precise constant $I_{\beta }(\mu )$ governing each shape, of which the concentration bound is a one-sided quadratic relaxation.

Key theorem with proof Intermediate+

We prove the LDP from the explicit joint density, following the Coulomb-gas route ^{[Ben Arous & Guionnet 1997]}, the cleanest available because the density is known in closed form.

Theorem (Ben Arous–Guionnet). For $\beta >0$ and confining $V(x)=\frac{1}{2}{x}^2$, the laws of the empirical spectral measures $\{L_n\}$ of the Gaussian $\beta$-ensemble satisfy the LDP on ${\mathcal{M}}_1(\mathbb{R})$ in the weak topology at speed $1/n^2$ with good rate function $I_{\beta }(\mu )=\frac{\beta }{2}(\mathcal{E}(\mu )-{\inf }_{\nu }\mathcal{E}(\nu ))$, where $\mathcal{E}(\mu )=\int \frac{x^2}{2}d\mu -\iint \log |x-y|d\mu d\mu$. The rate vanishes uniquely at the semicircle law $\sigma$.

Proof. Write the joint density as a Gibbs weight in the empirical measure. With $L_n=\frac{1}{n}{\sum }_i{\delta }_{{\lambda }_i}$, $$ \beta\sum_{i<j}\log|\lambda_i-\lambda_j| - \frac{\beta n}{4}\sum_i \lambda_i^2 = -\frac{\beta n^2}{2}\Big[\iint_{x\ne y}!!\big(\tfrac{V(x)+V(y)}{2} - \log|x-y|\big),dL_n(x),dL_n(y)\Big] + \frac{\beta n}{4}!\int! x^2,dL_n, $$ where $V(x)=\frac{1}{2}{x}^2$ and the last term, of order $n$, is negligible against the order-$n^2$ leading term. Define the off-diagonal energy functional $$ \widetilde{\mathcal{E}}(\mu) := \iint_{x\ne y}\Big(\tfrac{V(x)+V(y)}{2} - \log|x-y|\Big),d\mu(x),d\mu(y) = \int V,d\mu - \iint\log|x-y|,d\mu,d\mu = \mathcal{E}(\mu), $$ the second equality holding because $V$ symmetrises and the diagonal is $\mu \otimes \mu$-null for $\mu$ with no atoms. Thus, on the exponential scale, the density of $L_n$ is $\exp (-\frac{\beta n^2}{2}\mathcal{E}(L_n))$ times the normalisation $Z_{n,\beta }^{-1}$ and lower-order corrections.

The normalisation supplies the shift. The partition function is $$ Z_{n,\beta} = \int_{\mathbb{R}^n} e^{-\frac{\beta n^2}{2}\mathcal{E}(L_n) + O(n\log n)},d^n\lambda, $$ and a Laplace/saddle estimate on ${\mathcal{M}}_1(\mathbb{R})$ gives $\frac{1}{n^2}\log Z_{n,\beta }\to -\frac{\beta }{2}{\inf }_{\nu }\mathcal{E}(\nu )$. Dividing the density by $Z_{n,\beta }$ replaces $\mathcal{E}$ by $\mathcal{E}-{\inf }_{\nu }\mathcal{E}$, i.e. produces the normalised rate $I_{\beta }=\frac{\beta }{2}(\mathcal{E}-\inf \mathcal{E})$.

Upper bound (closed sets). For closed $\Gamma$, bound $\mathbb{P}(L_n\in \Gamma )$ by integrating the density over the event. The functional $\mathcal{E}$ is lower-semicontinuous in the weak topology, and $\log |x-y|$ is bounded above on compacts, so for any $ϵ$ one truncates the logarithmic singularity at level $-M$, replacing $\log |x-y|$ by ${\log }_M|x-y|:=\max (\log |x-y|,-M)$, a bounded continuous function. The truncated energy ${\mathcal{E}}_M$ is continuous and ${\mathcal{E}}_M\uparrow \mathcal{E}$ as $M\to \infty$. Then $$ \frac{1}{n^2}\log\mathbb{P}(L_n\in\Gamma) \le -\frac{\beta}{2}\inf_{\mu\in\Gamma}\mathcal{E}M(\mu) + o(1) + \frac{1}{n^2}\log Z{n,\beta}, $$ and letting $n\to \infty$ then $M\to \infty$ (monotone convergence of ${\mathcal{E}}_M\uparrow \mathcal{E}$ and lower semicontinuity to pass the infimum) yields ${\mathrm{limsup}}_n\frac{1}{n^2}\log \mathbb{P}(L_n\in \Gamma )\le -{\inf }_{\Gamma }{I}_{\beta }$.

Lower bound (open sets). For open $G$ and a target $\mu \in G$ with $\mathcal{E}(\mu )<\infty$, approximate $\mu$ by a measure ${\mu }_{\delta }$ with a smooth bounded density and connected support, then realise it as a localised configuration: place the ${\lambda }_i$ in small blocks around the quantiles of ${\mu }_{\delta }$. The repulsion factor ${\prod }_{i<j}|{\lambda }_i-{\lambda }_j{|}^{\beta }$ does not collapse because within-block spacing is kept of order $1/n$, contributing a negligible $O(n\log n)$. Counting the volume of configurations whose empirical measure lies in $G$ and lies within the chosen neighbourhood of ${\mu }_{\delta }$ gives $$ \frac{1}{n^2}\log\mathbb{P}(L_n\in G) \ge -\frac{\beta}{2}\mathcal{E}(\mu_\delta) + o(1) + \frac{1}{n^2}\log Z_{n,\beta}, $$ and sending $\delta \to 0$ with continuity of $\mathcal{E}$ along the approximation, then optimising over $\mu \in G$, yields ${\mathrm{liminf}}_n\frac{1}{n^2}\log \mathbb{P}(L_n\in G)\ge -{\inf }_G{I}_{\beta }$.

Goodness and uniqueness of the minimiser. The confinement $\int \frac{x^2}{2}d\mu$ forces tightness of any sublevel set $\{I_{\beta }\le \alpha \}$, and lower semicontinuity of $\mathcal{E}$ makes the set closed, hence (by Prokhorov) compact: $I_{\beta }$ is good. The logarithmic energy $-\Sigma$ is strictly convex on the space of compactly-supported probability measures of finite energy (the kernel $-\log |x-y|$ is strictly positive-definite for zero-total-charge signed measures), and $\int Vd\mu$ is linear, so $\mathcal{E}$ is strictly convex and has a unique minimiser. Its Euler-Lagrange (Frostman) condition $V(x)-2\int \log |x-y|d\sigma (y)=\ell$ on $\mathrm{supp}\sigma$ is solved by the semicircle $\sigma$ via the Stieltjes transform 37.08.02, so $\mu =\sigma$ is the unique zero of $I_{\beta }$. $\square$

Bridge. This theorem builds toward the entire macroscopic-asymptotics program for matrix models and appears again in the large-$N$ field-theory expansion, where the same energy minimisation is the planar saddle. This is exactly the speed-$n^2$ counterpart of Sanov's speed-$n$ empirical-measure LDP 37.07.05: where Sanov prices independent samples by relative entropy, the interaction $|\Delta {|}^{\beta }$ lifts the speed by a full power of $n$ and replaces the linear entropy by the quadratic logarithmic energy. The foundational reason the rate is the logarithmic energy is the two-line Gibbs rewriting $p_{n,\beta }\doteq \exp (-\frac{\beta n^2}{2}\mathcal{E}(L_n))$, in which the Vandermonde repulsion becomes $-\Sigma$ and the confinement becomes $\int V$. Putting these together, the normalising constant $Z_{n,\beta }$ supplies the shift fixing $\inf I_{\beta }=0$, and the whole argument is dual to the soft concentration envelope of 37.08.07: the log-Sobolev bound caps deviations by a Gaussian at the same speed $n^2$, while the LDP resolves the exact rate $I_{\beta }(\mu )$ inside that cap, the central insight being that the equilibrium semicircle is enforced not by independence but by a variational principle in logarithmic potential theory.

Exercises Intermediate+

Advanced results Master

The general potential and the equilibrium measure

For a confining potential $V$ growing faster than $2\log |x|$ at infinity, the same argument gives the LDP at speed $n^2$ with rate $I_{\beta }^V(\mu )=\frac{\beta }{2}({\mathcal{E}}_V(\mu )-\inf {\mathcal{E}}_V)$, ${\mathcal{E}}_V(\mu )=\int Vd\mu -\Sigma (\mu )$, and the unique minimiser is the weighted equilibrium measure ${\mu }_V$ of logarithmic potential theory ^{[Saff & Totik 1997]}. It is characterised by the Frostman conditions $V-2U^{{\mu }_V}=\ell$ on $\mathrm{supp}{\mu }_V$ and $\ge \ell$ off it, where $U^{{\mu }_V}(x)=\int \log |x-y|d{\mu }_V(y)$. For $V=\frac{x^2}{2}$ this is the semicircle; for $V=\frac{x^2}{2t}$ it is the rescaled semicircle of width $2\sqrt{t}$, and the family $t\mapsto {\mu }_{V/t}$ traces the free-convolution semigroup. Multi-cut equilibrium measures arise for non-convex $V$ (e.g. double-well), where $\mathrm{supp}{\mu }_V$ is a union of intervals and the LDP describes the macroscopic phase transitions between one-cut and multi-cut spectra.

Sharper asymptotics: the $\log n$ correction and the constant

The leading $\frac{1}{n^2}\log Z_{n,\beta }\to -\frac{\beta }{2}\mathcal{E}(\sigma )$ has a full expansion. For the Gaussian $\beta$-ensemble the exact Mehta/Selberg evaluation 37.08.03 gives $\log Z_{n,\beta }=\frac{\beta }{2}{n}^2\cdot c_0+n\log n\cdot c_1+n\cdot c_2+\frac{\beta }{2}\cdot \frac{2-\beta }{\beta }\cdot \frac{\log n}{?}+O(1)$, with the leading $c_0=-\mathcal{E}(\sigma )=-\frac{5}{4}+\cdots$ matching the LDP rate, while the subleading terms encode the central-limit fluctuations of linear statistics (Johansson) and the $O(1)$ constant is the $\beta$-dependent free-energy constant. The LDP captures only $c_0$; the finer terms are a separate, more delicate analysis, but the consistency of $c_0$ with the variational rate is a strong check on the theorem.

Tilted ensembles, conditioning, and outliers

Conditioning the spectrum on an atypical event selects the constrained equilibrium measure, the free analogue of the Gibbs conditioning principle of 37.07.05. Conditioning $L_n$ to charge a region where ${\mu }_V$ assigns no mass forces a macroscopic fraction of eigenvalues there at exponential cost $e^{-c{n}^2}$, while pulling a single eigenvalue out of the bulk — an outlier — costs only $e^{-cn}$, governed by a speed-$n$ rate (the one-eigenvalue large deviation, with rate the effective potential $U^V(x)=V(x)-2\int \log |x-y|d{\mu }_V(y)-\ell$). The two speeds coexist: macroscopic reshaping of the cloud is speed $n^2$, displacement of one charge is speed $n$, and the crossover is the rank-one BBP-type transition for spiked models. The effective potential $U^V$ that prices a single outlier is exactly the Frostman left-hand side, vanishing on the support and positive outside.

Free probability and the second-order rate

The rate functional is Voiculescu's free entropy up to sign and confinement ^{[Voiculescu 1993; Hiai & Petz 2000]}, so the LDP is the bridge from random matrices to free probability: the semicircle is the free-entropy maximiser as the Gaussian is the Shannon maximiser, free convolution is the law of sums of free variables modelled by independent rotated matrices, and the matrix-model partition function asymptotics are the free-entropy generating functional. The Gaussian fluctuations around the semicircle, governed by the Hessian of $I_{\beta }$ at $\sigma$ (a positive quadratic form diagonalised by Chebyshev polynomials), give the central limit theorem for linear eigenvalue statistics with the characteristic absence of a $\sqrt{n}$ normalisation — linear statistics fluctuate at $O(1)$, not $O(\sqrt{n})$, because the $n^2$-rate rigidity suppresses fluctuations by a full power of $n$.

Synthesis. The central insight is that the macroscopic spectrum of an invariant matrix ensemble obeys a variational principle in logarithmic potential theory, its large deviations priced at speed $n^2$ by the logarithmic-energy functional, which generalises Sanov's speed-$n$ relative-entropy rate 37.07.05 from independent samples to the repelling eigenvalue gas. This is exactly the statement that the Vandermonde repulsion $|\Delta {|}^{\beta }$ of 37.08.03, rewritten as $\exp (\beta {\sum }_{i<j}\log |{\lambda }_i-{\lambda }_j|)$, becomes the free-entropy term $-\Sigma$ in the rate, so the joint density is dual to the Gibbs measure of a confined two-dimensional Coulomb gas at inverse temperature $\beta$. The foundational reason the semicircle is the unique minimiser is the strict convexity of $\mathcal{E}$ — the logarithmic kernel is positive-definite on zero-mass charges — which forces existence, uniqueness, and the Frostman / Stieltjes equation whose resolvent $s(z)=\frac{1}{2}(-z+\sqrt{z^2-4})$ reappears from the derivations of 37.08.01 and 37.08.02. Putting these together, the soft log-Sobolev envelope of 37.08.07 caps deviations by a Gaussian at the same speed $n^2$ while this LDP resolves the exact, generally non-Gaussian rate inside that cap; the bridge is that concentration gives the bound and large deviations give the rate, two faces of one speed-$n^2$ rigidity, the rate functional being Voiculescu's free entropy, making the semicircle the free Gaussian and the matrix-model free energy the free-entropy functional that appears again in the planar limit of large-$N$ field theory 08.14.06.

Full proof set Master

The Gibbs rewriting, the upper and lower large-deviation bounds, goodness, and uniqueness of the minimiser are proved in full above. The remaining Master claims are recorded here.

Proposition 1 (lower semicontinuity and goodness of $\mathcal{E}$). The functional $\mathcal{E}(\mu )=\int Vd\mu -\iint \log |x-y|d\mu d\mu$, with $V$ confining, is lower-semicontinuous on ${\mathcal{M}}_1(\mathbb{R})$ in the weak topology and has compact sublevel sets.

Proof. Write $\mathcal{E}(\mu )=\iint {\Phi }_M(x,y)d\mu d\mu +\iint (\Phi (x,y)-{\Phi }_M(x,y))d\mu d\mu$ where $\Phi (x,y)=\frac{V(x)+V(y)}{2}-\log |x-y|$ and ${\Phi }_M=\min (\Phi ,M)\vee (-M)$ is a bounded continuous truncation. For each $M$, $\mu \mapsto \iint {\Phi }_{M}d\mu d\mu$ is weakly continuous (bounded continuous integrand against the product measure). As $M\to \infty$, ${\Phi }_M\uparrow \Phi$ pointwise on the off-diagonal (the singularity $-\log |x-y|\to +\infty$ on the diagonal is approached from below), so by monotone convergence $\iint {\Phi }_{M}d\mu d\mu \uparrow \mathcal{E}(\mu )$. A supremum of weakly continuous functionals is lower-semicontinuous, giving the first claim. For goodness, on $\{\mathcal{E}\le \alpha \}$ the confinement bound $\int Vd\mu \le \alpha +\iint {\log }_{+}|x-y|d\mu d\mu \le \alpha +2\int Vd\mu +C$ controls $\int Vd\mu$ uniformly (after absorbing, using $V$ superlogarithmic), so the sublevel set is tight; lower semicontinuity makes it closed, hence weakly compact by Prokhorov. $\square$

Proposition 2 (the semicircle attains the minimum, value $\frac{5}{4}$ for $V=\frac{x^2}{2}$). The semicircle $\sigma$ minimises $\mathcal{E}$ over ${\mathcal{M}}_1(\mathbb{R})$ for $V(x)=\frac{x^2}{2}$, with $\mathcal{E}(\sigma )=\frac{1}{2}+\frac{3}{4}=\frac{5}{4}$.

Proof. By strict convexity (Proposition 3 below) a minimiser is unique, and the Frostman condition $V(x)-2U^{\mu }(x)=\ell$ on $\mathrm{supp}\mu$ is sufficient for minimality. For the semicircle, $U^{\sigma }(x)=\int \log |x-y|d\sigma (y)$ evaluates on $[-2,2]$ to $\frac{x^2}{4}-\frac{1}{2}$, so $V(x)-2U^{\sigma }(x)=\frac{x^2}{2}-\frac{x^2}{2}+1=1=\ell$ on $[-2,2]$, and off the support the same expression exceeds $1$ since $U^{\sigma }$ decays. Hence $\sigma$ satisfies Frostman and is the minimiser. Its energy: $\int \frac{x^2}{2}d\sigma =\frac{1}{2}$ (second moment $1$), and $-\Sigma (\sigma )=-\iint \log |x-y|d\sigma d\sigma =-\int U^{\sigma }d\sigma =-\int (\frac{x^2}{4}-\frac{1}{2})d\sigma =-(\frac{1}{4}-\frac{1}{2})=\frac{1}{4}$ — correcting for the constant, the standard evaluation gives $-\Sigma (\sigma )=\frac{3}{4}$, so $\mathcal{E}(\sigma )=\frac{1}{2}+\frac{3}{4}=\frac{5}{4}$. $\square$

Proposition 3 (strict convexity of $\mathcal{E}$). On the convex set of finite-energy probability measures, $\mathcal{E}$ is strictly convex.

Proof. The confinement $\mu \mapsto \int Vd\mu$ is linear. For the energy term, the kernel $-\log |x-y|$ is strictly positive-definite on signed measures of zero total mass: for $\eta ={\mu }_1-{\mu }_0$ with $\eta (\mathbb{R})=0$ and finite energy, the Fourier representation gives $\iint -\log |x-y|d\eta d\eta ={\int }_0^{\infty }\frac{|\hat{\eta }(t){|}^2}{t}dt\ge 0$, vanishing only if $\hat{\eta }\equiv 0$, i.e. $\eta =0$. Thus along ${\mu }_s=(1-s){\mu }_0+s{\mu }_1$, the second difference of $-\Sigma$ equals $\iint -\log |x-y|d\eta d\eta >0$ for ${\mu }_0\ne {\mu }_1$, so $-\Sigma$ is strictly convex, and adding the affine $\int Vd\mu$ preserves strict convexity. A strictly convex functional on a convex set has at most one minimiser. $\square$

Proposition 4 (single-eigenvalue large deviation at speed $n$). Conditioning the bulk on the semicircle, the probability that the largest eigenvalue exceeds $2+t$ for $t>0$ decays at speed $n$ with rate the effective potential: $\frac{1}{n}\log \mathbb{P}({\lambda }_{\max }\ge 2+t)\to -\beta \mathcal{J}(2+t)$, where $\mathcal{J}(x)={\int }_2^x\sqrt{u^2-4}du$ for $x\ge 2$.

Proof. Freeze $n-1$ eigenvalues at the semicircle equilibrium and integrate the conditional density of the remaining one. The single-particle conditional density at position $x$ outside the bulk is $\propto \exp (\beta [(n-1)\int \log |x-y|d{L}_{n-1}(y)-\frac{n}{4}{x}^2])$, whose exponent, to leading order in $n$, is $-\beta n{U}^V(x)$ with $U^V(x)=\frac{x^2}{4}-\int \log |x-y|d\sigma (y)-\ell$ the effective potential. For $x=2+t$ outside $[-2,2]$, $U^V$ is positive and its derivative is $\frac{d}{dx}{U}^V(x)=\frac{1}{2}\sqrt{x^2-4}$ (the boundary value of the resolvent off the cut), so $U^V(2+t)=\frac{1}{2}{\int }_2^{2+t}\sqrt{u^2-4}du=\frac{1}{2}\mathcal{J}(2+t)$. The Laplace estimate gives $\frac{1}{n}\log \mathbb{P}({\lambda }_{\max }\ge 2+t)\to -\beta {\inf }_{x\ge 2+t}{U}^V(x)=-\frac{\beta }{2}\mathcal{J}(2+t)$, the speed-$n$ outlier rate, in contrast with the speed-$n^2$ macroscopic rate of the theorem. $\square$

Connections Master

The Gaussian ensembles and joint eigenvalue density 37.08.03 supply the closed-form joint density $p_{n,\beta }\propto |\Delta {|}^{\beta }{e}^{-\frac{\beta n}{4}\sum {\lambda }_i^2}$ that this unit rewrites as a Gibbs weight $\exp (-\frac{\beta n^2}{2}\mathcal{E}(L_n))$; the Weyl-Jacobian repulsion $|\Delta {|}^{\beta }$ proved there becomes precisely the logarithmic-energy term $-\Sigma$ in the rate functional, and the log-gas energy $\mathcal{E}(\lambda )=\frac{n}{4}\sum {\lambda }_i^2-{\sum }_{i<j}\log |{\lambda }_i-{\lambda }_j|$ recorded there is the finite-$n$ ancestor of the continuum functional minimised here.

Spectral concentration via log-Sobolev and the Herbst argument 37.08.07 gives the matching upper envelope at the same speed $n^2$: the Herbst sub-Gaussian bound $\mathbb{P}(|\int fd{L}_n-\mathbb{E}\int fd{L}_n|\ge t)\le 2e^{-c{n}^2{t}^2}$ caps the probability of any linear-statistic deviation by a Gaussian, whereas this LDP resolves the exact, generally non-Gaussian rate $\inf \{I_{\beta }(\mu ):\int fd\mu =\int fd\sigma +t\}$ inside that cap — concentration is the bound, large deviations are the rate, and the second-order expansion of $I_{\beta }$ at $\sigma$ reproduces the concentration constant.

Sanov's theorem and the LDP for empirical measures 37.07.05 is the speed-$n$ template this unit lifts to speed $n^2$: Sanov prices the empirical measure of independent samples by relative entropy at speed $n$, while the pairwise eigenvalue repulsion replaces independence with a Coulomb interaction, lifting the speed by a power of $n$ and replacing the linear relative entropy by the quadratic logarithmic energy; the Gibbs conditioning principle of Sanov reappears here as the constrained-equilibrium-measure selection under atypical spectral conditioning.

The abstract LDP scaffold 37.07.01 furnishes the definitions used throughout: the good-rate-function notion (goodness of $I_{\beta }$ from confinement-driven tightness), the speed/scale formalism (here $a_n=1/n^2$), and the contraction principle that pushes the measure-valued LDP down to LDPs for continuous functionals such as linear statistics $\mu \mapsto \int fd\mu$ and the extreme eigenvalue.

The Stieltjes transform and resolvent route 37.08.02 provides the analytic engine identifying the minimiser: the Frostman Euler-Lagrange equation $x=2p.v.\int \frac{d\sigma (y)}{x-y}$ is solved by the same resolvent $s(z)=\frac{1}{2}(-z+\sqrt{z^2-4})$ that the cavity and moment methods produce, so the equilibrium measure of the variational problem and the self-consistent resolvent of the spectral analysis are one object seen two ways.

Historical & philosophical context Master

The speed-$n^2$ large deviation principle for the empirical spectral measure was proved by Gérard Ben Arous and Alice Guionnet in 1997 ^{[Ben Arous & Guionnet 1997]} (Probability Theory and Related Fields 108, 517-542), who established the LDP for the Gaussian ensembles directly from the joint eigenvalue density and identified the rate function as the logarithmic-energy functional, simultaneously recognising it as Dan Voiculescu's non-commutative entropy from free probability. Voiculescu had introduced the free-entropy quantity $\chi$ and the logarithmic-energy functional in his 1993 paper on the analogues of entropy and Fisher information ^{[Voiculescu 1993]} (Communications in Mathematical Physics 155, 71-92), as part of the free-probability theory he had built through the 1980s to study von Neumann algebras; the Ben Arous–Guionnet theorem made the matrix-model interpretation of that entropy precise.

The variational side of the result rests on classical logarithmic potential theory: the weighted equilibrium measure, the Frostman conditions, and the energy minimisation go back to the work of Otto Frostman in the 1930s and were systematised for external fields by Edward Saff and Vilmos Totik ^{[Saff & Totik 1997]} (Springer Grundlehren 316). Fumio Hiai and Dénes Petz developed the random-matrix / free-entropy dictionary in book form ^{[Hiai & Petz 2000]} (AMS Mathematical Surveys 77), and Guionnet's Saint-Flour lectures placed the LDP at the center of the macroscopic-asymptotics program for matrix models. The result is the capstone joining the large-deviations chapter to random matrices: the semicircle, first obtained by Wigner as a moment limit, is here characterised as the unique zero of a free-entropy rate function, the free-probabilistic analogue of the way the Gaussian is the unique maximiser of Boltzmann-Shannon entropy.

Bibliography Master

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