Bulk Universality of General $β$-Ensembles with Non-convex Potential

Introduction and the main results

The universality conjecture asserts that the eigenvalue gap distributions in the bulk depend only on $\beta$ and are independent of the potential $V$. For classical ensembles, the eigenvalue correlation functions can be explicitly expressed in terms of polynomials orthogonal to the measure $e^{-\beta V(x)/2}$. Thus the analysis of the correlation functions relies heavily on the asymptotic properties of the corresponding orthogonal polynomials. This approach, initiated by Dyson, Gaudin and Mehta (see [Meh1991] for a review) was the starting point for all results on classical universality. Precise analysis on orthogonal polynomials for general class of weight functions was made possible by the Riemann-Hilbert approach [BleIts1999, DeiKriMcLVenZho1999I, DeiKriMcLVenZho1999II]. There are also methods independent of the Riemann-Hilbert approach, see, e.g., [PasShc1997, PasShc2008, Lub2009]. The universality for $\beta=2$ was proved for very general potential. For $\beta=1,4$ [DeiGio2009, KriShc2011, Shc2011] it was proved for analytic $V$ with some additional conditions. A summary of recent developments can be found in [AndGuiZei2010, Dei1999, DeiGio2009, PasShc2011].

For non-classical values of $\beta$, i.e., $\beta\not\in\{1,2,4\}$, there is no simple expression of the correlation functions in terms of orthogonal polynomials. In [BouErdYau2011], we initiated a new approach to prove bulk universality for all $\beta>0$ and strictly convex $V$. The method was based on estimating correlation functions by local Dirichlet form and the main ingredients consist of the following two steps:

Step 1. Rigidity of eigenvalues. This establishes that the location of the eigenvalues are not too far from their classical locations determined by the equilibrium density $\rho(s)$.

Step 2. Uniqueness of local Gibbs measures with logarithmic interactions. With the precision of eigenvalue location estimates from Step 1 as an input, the eigenvalue gap distributions are shown to be given by the corresponding Gaussian ones. (We will take the uniqueness of the gap distributions as our definition of the uniqueness of Gibbs state.)

Our goal is to extend this result to the non-convex case. It was emphasized in [BouErdYau2011] that the convexity of the potential $V$ was used only in Step 1. So in order to apply this method, it suffices to prove the rigidity estimate which we now introduce.

for some $\alpha>0$, if $|x|$ is large enough. It is known [BouPasShc1995] that under these (in fact, even weaker) conditions the measure is normalizable, $Z_{N}<\infty$. Moreover, the averaged density of the empirical spectral measure, defined as

converges weakly to a continuous function $\rho$, the equilibrium density, with compact support. We additionally assume that $\rho(s)$ is supported on a single interval $[A,B]$, and that $V$ is regular in the sense of [KuiMcL2000]. We recall that $V$ is regular if its equilibrium density $\rho$ is positive on $(A,B)$ and vanishes like a square rootThis is not a strong constraint: [KuiMcL2000] proves that the regular potentials $V$ are a dense and open subset of the potentials for the topology induced by the distance $d(V,W)=\sum_{j=0}^{3}\sum_{k=1}^{\infty}2^{-k}\frac{\|V^{(j)}-W^{(j)}\|_{\mbox{L}^{\infty}[-k,k]}}{1+\|V^{(j)}-W^{(j)}\|_{\mbox{L}^{\infty}[-k,k]}}+\sum_{k=1}^{\infty}2^{-k}\frac{|G_{k}(V)-G_{k}(W)|}{1+|G_{k}(V)-G_{k}(W)|},$ where $G_{k}(V)=\inf_{|x|>k}V(x)/\log|x|$. at each of the endpoints of $[A,B]$, that is

In this paper, we are interested in the usual $n$-point correlation functions, generalizing $\rho_{1}^{(N)}$, and defined by

where $\lambda^{(\sigma)}=(\lambda_{\sigma(1)},\dots,\lambda_{\sigma(N)})$, with $\lambda_{\sigma(1)}<\dots<\lambda_{\sigma(N)}$.

It is known that the particles are rigid, i.e. they cannot be far from their classical locations For eigenvalues in the bulk, (1.7) follows from the large deviations for the empirical spectral measure with speed $N^{2}$ [BenGui1997, AndGuiZei2010], and for the extreme eigenvalues the large deviations principle with speed $N$ was proved in [AndGuiZei2010], Theorem 2.6.6, up to a condition on the partition function that follows from Theorem 1 (iii) in [Shc2011].: for any $\varepsilon>0$ there are positive constants $c_{1}$, $c_{2}$ such that, for all $N\geq 1$,

The main technical result of this paper is to prove that rigidity holds for the measure $\mu$ at the optimal scale $1/N$ in the bulk in the following sense. This theorem extends our rigidity result in [BouErdYau2011] to non-convex potential $V$.

Let $V$ be real analytic, regular with equilibrium density supported on a single interval $[A,B]$, and satisfy (1.2), (1.3). Take any $\alpha>0$ and $\varepsilon>0$. Then there are constants $\delta,c_{1},c_{2}>0$ such that for any $N\geq 1$ and $k\in\llbracket\alpha N,(1-\alpha)N\rrbracket$,

Our main result on the universality is the following theorem:

Let $V$ be real analytic, regular with equilibrium density supported on a single interval $[A,B]$, and satisfy (1.2), (1.3). Then for any $\beta>0$ the bulk universality holds for the $\beta$-ensemble $\mu=\mu_{\beta,V}$. More precisely, for any $E\in(A,B)$ and $|E^{\prime}|<2$, for any smooth test functions $O$ with compact support and for any $0<k\leq\frac{1}{2}$, we have, with $s:=N^{-1+k}$, that

Here $\rho_{sc}(E)=\frac{1}{2\pi}\sqrt{4-E^{2}}$ is the Wigner semicircle law and $\rho_{{\rm Gauss},n}^{(N)}$ are the correlation functions of the Gaussian $\beta$-ensemble, i.e. with $V(x)=x^{2}$.

Theorem 1.2 follows immediately from the rigidity estimates, (1.7), (1.8), and the uniqueness of local Gibbs measure, i.e., Theorem 2.1 and Corollary 2.2 in [BouErdYau2011]. We note that the proof of the latter results in Section 4 of [BouErdYau2011] uses only the rigidity estimate, given in Theorem 3.1 of [BouErdYau2011], as an input. Once the rigidity estimate is proven, the rest of the argument is identical and we will not repeat it here.

The rest of this paper is devoted to the proof of Theorem 1.1. After some initial estimates concerning the large deviations regime and global smooth linear statistics (Section 2), the proof consists in the following steps. First we compare $\mu$ to some convexified measures $\nu$ (Section 3); the Hamiltonian ${\mathcal{H}}_{\nu}$ of $\nu$ differs from that of $\mu$ mainly by some properly chosen linear statistics of the $\lambda_{i}$’s, allowing ${\mathcal{H}}_{\nu}$ to be convex. Despite this change in convexity, we will prove that the two measures $\mu$ and $\nu$ have the same subexponentially small probability events. This step is the main extra ingredient allowing one to generalize the rigidity estimate obtained in [BouErdYau2011]. Then by a self-improving method, this measure $\nu$ (together with $\mu$) is proved to have rigidity till the optimal scale, thanks to comparisons with locally constrained versions of $\nu$ (Section 4).

Preliminary results

if $\int V(t){\rm d}\nu(t)<\infty$, and $I(\nu)=\infty$ otherwise. Moreover, if one assumes that $\rho$ is supported on a single interval $[A,B]$ and regular in the sense of the previous section, $\rho$ has the following properties:

In order to have the density supported strictly in a compact interval, for given $\kappa>0$, define the following variant of $\mu^{(N)}$ conditioned to have all particles in $[A-\kappa,B+\kappa]$:

In this paper we will choose $\kappa$ to be small. This choice differs from [BouErdYau2011] where, instead of $[A-\kappa,B+\kappa]$, we restricted the particles to $[-R,R]$ for a very large $R$. The smaller interval is needed here because we need $r$ to be positive on the support of $\mu^{(N,\kappa)}$ in the proof of Lemma 2.2. Unlike in the case of convex $V$ where $r$ is known to have no real zero at all, for the non-convex regular case we only know that $r$ is nonzero in the interval $[A,B]$. By continuity, it is also nonzero in $[A-\kappa,B+\kappa]$ for some small $\kappa$.

Let $\rho_{k}^{(N,\kappa)}$ denote the correlation functions of the measure $\mu^{(N,\kappa)}$. Then Lemma 1 in [BouPasShc1995] states that under condition (1.3), for some large enough $\kappa$ there exists some $c>0$, depending only on $V$, such that for any $x_{1},\dots,x_{k}\in[A-\kappa,B+\kappa]$, we have

and for $x_{1},\dots,x_{j}\not\in[A-\kappa,B+\kappa]$, $x_{j+1},\dots,x_{k}\in[A-\kappa,B+\kappa]$,

The estimates (2.5) and (2.6) actually also hold for arbitrarily small fixed $\kappa>0$ thanks to the large deviations estimates (1.7).

2 Linear statistics

The following lemma was essentially proven in [Shc2011] (for the variance of linear statistics).

For any function $\phi$ with $\|\phi\|_{\infty}+\|\phi^{\prime}\|_{\infty}+\|\phi^{\prime\prime}\|_{\infty}<\infty$, there is a constant $c>0$ depending only on $V$ and $\phi$ (one can choose $c=\operatorname{O}(\|\phi\|_{\infty}+\|\phi^{\prime}\|_{\infty}+\|\phi^{\prime\prime}\|_{\infty})$) such that, for any $N\geq 1$ and $s>0$,

Proof. Without loss of generality, we can assume that $\phi$ is compactly supported (thanks to large deviation estimates such as (2.6)). We know from Shcherbina, equation (2.22) in [Shc2011], that for the Stieltjes transforms, i.e. $g(u)=1/(z-u)$, there is a constant $c>0$ depending only on $V$ and $g$ (one can choose $c=\operatorname{O}(\|g^{(4)}\|_{\infty})$) such that, for any $N\geq 1$,

where $\mu_{h}$ is obtained by replacing $V$ by $V+\frac{h}{N}$ in the definition of $\mu$, and $h$ is for example any $N$-independent smooth compactly supported function. We will now prove that this implies that (2.7) actually holds when replacing $g$ by any smooth compactly supported $\phi$, for example by a Helffer-Sjöstrand type argument, similar to Lemma 2.3. We can now apply formula (B.13) in [ErdRamSchYau2010] for the signed measure $\widetilde{\rho}=\rho^{(N,\mu_{h})}_{1}-\rho$, with Stieltjes transform $S$, where $\rho^{(N,\mu_{h})}_{1}$ is the one-point correlation function of $\mu_{h}$. We obtain

Following now Lemma 1 in [Shc2011], consider

Then obviously $\frac{{\rm d}^{2}}{{\rm d}t^{2}}\log Z_{N}(t)\geq 0$, so

so using (2.7) we get that $Z_{N}(t)\leq e^{c|t|}$, from which Lemma 2.1 easily follows.

3 Analysis of the loop equation

This section analyzes the loop equation (2.10) in the following Lemma 2.2. Its proof is very similar to [BouErdYau2011] except that, instead of the logarithmic Sobolev inequality which was valid only for convex $V$, we will use Lemma 2.1. Furthermore, since the support of the restricted measure $\mu^{(N,\kappa)}$ has changed, the integration contours in (2.16) are chosen slightly differently from those in [BouErdYau2011].

In the form presented here, we follow closely the proof in [Shc2011]. We now introduce some notations needed in the proof.

$m_{N}$ is the Stieltjes transform of $\rho^{(N)}_{1}(s){\rm d}s$, evaluated at some $z$ with $\operatorname{Im}(z)>0$, and $m$ its limit:

$s(z)=-2r(z)\sqrt{(A-z)(B-z)}$, where the square root is defined such that

finally, $c_{N}(z)=\frac{1}{N^{2}}k_{N}(z)+\frac{1}{N}\left(\frac{2}{\beta}-1\right)m_{N}^{\prime}(z)$, where

The loop equation (see [Joh1998, Eyn2003, Shc2011] for various proofs) is

In the regime where $|m_{N}-m|$ is small, we can neglect the quadratic term. The term $b_{N}$ is the same order as $|m_{N}-m|$ and is difficult to treat. As observed in [AlbPasShc2001, Shc2011], for analytic $V$ (hence analytic $b_{N}$), this term vanishes when we perform a contour integration. So we have roughly the relation

where we dropped the less important error involving $m_{N}^{\prime}(z)/N$ due to the extra $1/N$ factor. With no convexity assumption on $V$, the difficulty will be to estimate the above variance to immediately obtain an estimate on $m_{N}-m$; this is the reason why we will introduce a convexified version of the measure $\mu$ in the next Section 3. To quantify more precisely (2.11) we will use the following result, already proved in [BouErdYau2011] for convex $V$.

as $N\to\infty$ uniformly in $\eta\geq N^{-1+a}$ for some $0<a<1$. Then there are constants $c,\kappa>0$ such that for any $N^{-1+a}\leq\eta\leq\kappa$, $A+\delta<E<B-\delta$,

Proof. First, for technical contour integration reasons, it will be easier to consider the measure (2.4) instead of $\mu^{(N)}$ here. More precisely, define

Then it is a direct consequence of (2.5) and (2.6) that for any $\kappa>0$ there is a constant $c>0$ such that uniformly on $\eta\geq N^{-10}$ (or any power of $N$),

Note that the above expression makes sense for large enough $N$, because then $r$ has no zero on $\mathcal{L}$. Using (2.14), this implies, for $\eta\geq N^{-1}$,

Now, as $\rho_{1}^{(N,\kappa)}$ and $\rho$ are supported on $[A-\kappa,B+\kappa]$, $m_{N}^{(\kappa)}-m$ and $c_{N}^{(\kappa)}$ are uniformly $\operatorname{O}(1)$ in the vertical segments of $\mathcal{L}$. Consequently, from the above equation

As $b_{N}$ and $r$ are analytic inside $\mathcal{L}$, for $z$ outside $\mathcal{L}$ we get

Remember we define $f(z)=\sqrt{(A-z)(B-z)}$ uniquely by $f(z)\sim z$ as $z\to\infty$. Moreover, $|m_{N}^{(\kappa)}-m|(z)=\operatorname{O}(z^{-2})$ as $|z|\to\infty$ because $\rho$ and $\rho_{1}^{(N,\kappa)}$ are compactly supported:

Consequently, the function $s(m^{(\kappa)}_{N}-m)/r=-2f(m^{(\kappa)}_{N}-m)$ is $\operatorname{O}(z^{-1})$ as $|z|\to\infty$. Moreover, it is analytic outside $\mathcal{L}$, so the Cauchy integral formula yields

Consider now the following rectangular contours, defined by their vertices:

In particular, note that all the zeros of $r$ are strictly outside $\mathcal{L}_{2}$. For $z$ inside $\mathcal{L}_{2}$ and $\operatorname{Im}(z)\geq N^{-1}$, by the Cauchy formula, equation (2.15) implies that

In the above expression, if now $z$ is on $\mathcal{L}_{1}$, $|z-\xi|\geq\kappa$, and on $\mathcal{L}_{2}$ $|r|$ is separated away from zero by a positive universal constant. Moreover, $c_{N}^{(\kappa)}(\xi)$ can be bounded in the following way. For any $\xi\in\mathcal{L}_{2}$, there is a smooth function $g_{\xi}$ supported on $[A-2\kappa,B+2\kappa]$ which coincides with $\frac{1}{\xi-\lambda_{k}}$ on $[A-\kappa,B+\kappa]$, Moreover, this choice can be made such that $\|g_{\xi}\|_{\infty},\|g^{\prime}_{\xi}\|_{\infty},\|g^{\prime\prime}_{\xi}\|_{\infty}$ are uniformly bounded in $\xi\in\mathcal{L}_{2}$. Then

where the last equality follows from (2.5). Now, from Lemma 2.1, this last variance is uniformly bounded by $c\,(\log N)^{2}$, with $c$ uniformly bounded in $\xi$. This proves that $k^{(\kappa)}_{N}(\xi)$ is $\operatorname{O}((\log N)^{2}/N^{2})$, uniformly on the contour $\mathcal{L}_{2}$. Moreover, $\frac{1}{N}{m^{(\kappa)}_{N}}^{\prime}=\operatorname{O}(N^{-1})$, so finally $c_{N}^{(\kappa)}(\xi)$ is uniformly $\operatorname{O}(N^{-1})$ on $\mathcal{L}_{2}$ and (2.17) implies

Moreover, from the maximum principle for analytic functions, $\sup_{\mathcal{L}_{2}}|m_{N}^{(\kappa)}-m|\leq\sup_{\mathcal{L}_{1}}|m_{N}^{(\kappa)}-m|$, so the previous equation implies

We know that $\rho_{1}^{(N)}(s){\rm d}s$ converges weakly to $\rho(s){\rm d}s$ (see [AndGuiZei2010]), so by (2.5) and (2.6) $\rho_{1}^{(N,\kappa)}(s){\rm d}s$ converges weakly to $\rho(s){\rm d}s$. On $\mathcal{L}_{1}$, $z$ is at distance at least $\kappa$ from the support of both $\rho_{1}^{(N,\kappa)}(s){\rm d}s$ and $\rho(s){\rm d}s$ so, on $\mathcal{L}_{1}$, $m_{N}^{(\kappa)}-m$ converges uniformly to . Together with the above equation, this implies that

By the maximum principle the same estimate holds outside $\mathcal{L}_{1}$, in particular on $\mathcal{L}_{2}$, so equation (2.17) implies that for $z$ inside $\mathcal{L}_{1}$

for some constant $c$. We used the well-known fact that $\operatorname{Im}\ m$ is uniformly bounded on the upper half planeThis follows for example from properties of the Cauchy operator, see p 183 in [Dei1999].. On the set $A+\delta<E<B-\delta$ and $|\eta|<\kappa$, we have $\inf|s|>0$. Therefore (2.18) takes the form

From the hypothesis (2.12), if $N^{-1+a}\leq\eta\leq\kappa$ and $A+\delta<E<B-\delta$, then

The same conclusion remains when substituting $m_{N}^{(\kappa)}$ (resp. $k_{N}^{(\kappa)}$) by $m_{N}$ (resp. $k_{N}$) thanks to (2.5) and (2.6).

To prove rigidity results for $\mu$, the above Lemma 2.2 will be combined with the following Helffer-Sjöstrand estimate, already proved in the following form in [BouErdYau2011].

Then for some constant $C>0$, independent of $N$ and $E\in[A+\delta,B-\delta]$, we have

Convexification

The Hamiltonian ${\mathcal{H}}={\mathcal{H}}_{N}$ of the measure $\mu\sim\exp(-\beta N{\mathcal{H}})$ is given by

${\mathcal{H}}$ is not convex, but its Hessian is bounded from below, $\nabla^{2}{\mathcal{H}}\geq-W$. We will modify this Hamiltonian by an additional term

Compared with $\gamma_{j}$ defined in (1.6), there is a small shift in the definition which makes a technical step (Lemma 3.3) easier in this section. In all estimates involving $\gamma_{j}$ this small shift plays no role since $\max_{j}|\gamma_{j}-{\widetilde{\gamma}}_{j}|\leq CN^{-2/3}$. In particular the crude large deviation bound (1.7) holds for ${\widetilde{\gamma}}$’s as well:

The $N^{-1/2}$ normalization in the definition of $X_{\alpha}$ is chosen such that the vector

where we used a simple Schwarz inequality

such that for typical point configuration \mbox{\boldmath\lambda}=(\lambda_{1},\lambda_{2},\ldots,\lambda_{N}) we have

2 Slow modes analysis

Let $\varepsilon>0$ be sufficiently small, depending only on $V$. Define

Then there is a constant $c_{1}>0$ depending only on $V$ such that for any $\varepsilon>0$ there is a constant $c_{2}>0$ (depending on $V$ and $\varepsilon$) such that for any $N$ and $i,j\in\llbracket 1,N\rrbracket$

The relation between $Q_{i,j}$ and $R_{i,j}$ is dictated by the requirement that

Proof. Recall that $[A,B]$ is the support of $\rho$, $\rho>0$ on $(A,B)$ and $\rho$ has a square-root singularity at the two endpoints, i.e. it vanishes as $\rho(x)\sim s_{A}\sqrt{x-A}$ as $x\to A^{+}$ and $\rho(x)\sim s_{B}\sqrt{B-x}$ as $x\to B^{-}$ with some positive $s_{A},s_{B}$.

From the large deviations of the extreme eigenvalues (included in (1.7)), we know that for any $\kappa>0$ there is a $c(\kappa)>0$ such that

Fix a positive number $s<\min(s_{A},s_{B})$. Then there is a $\kappa_{0}>0$, depending only on $V$, such that

Let $\varepsilon\leq c\kappa_{0}^{3/2}$ with a small positive constant $c$. Suppose that $k\leq N/2$; if $k$ is near the upper edge, the argument is similar. Since

with some positive constants $c,C$, depending only on $V$. Subtracting the first and second relations and using (3.3), we obtain that for any fixed $K$

apart from an event of exponentially small probability (i.e. of type $\exp(-c(\varepsilon/K)N)$).

Additionally, assume now that $k\geq N\varepsilon$. Under (3.15) we easily see that $\lambda_{k}\in({\widetilde{\gamma}}_{k/2},{\widetilde{\gamma}}_{3k/2})$, since both $\int_{{\widetilde{\gamma}}_{k/2}}^{{\widetilde{\gamma}}_{k}}\rho$ and $\int_{{\widetilde{\gamma}}_{k}}^{{\widetilde{\gamma}}_{3k/2}}\rho$ are of the order $k/N$ which is larger than $\varepsilon/K$ if $K$ is large enough (depending only on $V$). Then (3.13) and (3.15) imply

with exponentially high probability and with a constant $C$ depending only on $V$.

Now we consider the $k\leq N\varepsilon$ case. Using (3.12) with $\kappa=\varepsilon^{2/3}$ and (3.14), we have (apart from an event of exponentially small probability)

Finally, still when $k\leq N\varepsilon$, i.e. ${\widetilde{\gamma}}_{k}\leq A+C\varepsilon^{2/3}$ then (3.15) implies that $\lambda_{k}\leq A+C_{1}\varepsilon^{2/3}$ with a large $C_{1}$, i.e.

still apart from an event of exponentially small probability. Summarizing all cases, we obtain that

since either $i$ or $j$ is larger than $N\varepsilon$ and smaller than $N(1-\varepsilon)$, say $N\varepsilon\leq i\leq N(1-\varepsilon)$, and then $\rho$ is at least of order $\varepsilon^{1/3}$ in the neighborhood of ${\widetilde{\gamma}}_{i}$. If $|i-j|\leq N\varepsilon$, then we have the trivial bound $|{\widetilde{\gamma}}_{i}-{\widetilde{\gamma}}_{j}|\leq C\varepsilon^{2/3}$. Combining these,

holds for any $i,j$. Furthermore, clearly $|\lambda_{i}-{\widetilde{\gamma}}_{i}|\leq C\varepsilon^{2/3}$ from (3.16), so we have proved that

with overwhelming probability and for any $i,j$. In other words, there is a constant $C$ (depending only on $V$) such that for any sufficiently small $\varepsilon$ and for some $c(\varepsilon)>0$ we have for any $N$ and $i,j\in\llbracket 1,N\rrbracket$

The proof of Lemma 3.1 will therefore be complete if we can prove that

We use the matrix $Q$ in the previous lemma instead of bounds of type $(\ref{eqn:nonCirc})$ because it is related to $R$, a circulant matrix, allowing to derive its eigenvalues and eigenvectors in an explicit way.

In particular, for any given $W>0$ there is a sufficiently small $\varepsilon$ such that for large enough $N$ we have $\nu_{2N}>W$. Moreover, for any given $\varepsilon>0$ and $s>0$ there is some $a>0$ depending only on $\varepsilon$ and $s$ such that for any $N$

Proof. The first assertions, about the eigenvalues and eigenvectors, is a general fact about circulant matrices and can be obtained by Fourier transform in $\left\{0,\frac{1}{2N},\dots,\frac{2N-1}{2N}\right\}$.

Concerning the distribution of eigenvalues, note that

We therefore have, for sufficiently small $\varepsilon>0$, $\nu_{2N}>W$ for large enough $N$. We now write

where ${\mathcal{Q}}={\mathcal{Q}}^{(\varepsilon)}$ was defined in (3.5) with coefficients $Q_{i,j}=Q_{i,j}^{(\varepsilon)}$ defined in (3.11).

we just need to prove that the operator inequality

3 The locally constrained measures

In this section some arbitrary $\varepsilon,\alpha>0$ are fixed. Let $\theta$ be a continuous nonnegative function with $\theta=0$ on $$and$\theta^{\prime\prime}\geq 1$for$|x|>1$. We can take for example$\theta(x)=(x-1)^{2}\mathds{1}_{x>1}+(x+1)^{2}\mathds{1}_{x<-1}$ in the following.

the function $g_{\alpha}$ was defined in Lemma 3.3;

$X_{\alpha}=N^{-1/2}\sum_{j}\left(g_{\alpha}(\lambda_{j})-g_{\alpha}({\widetilde{\gamma}}_{j})\right)$;

$\psi^{(s)}(\lambda)=N\theta\left(\frac{s}{N}\sum_{i=1}^{N}(\lambda_{i}-{\widetilde{\gamma}}_{i})^{2}\right)$;

$\psi_{i,j}(\lambda)=\frac{1}{N}\theta\left(\sqrt{c_{1}\,N\,Q_{i,j}}(\lambda_{i}-\lambda_{j})\right)$, where $c_{1}$ was defined in Lemma 3.1.

4 Equivalence of the measures ν𝜈\nu and μ𝜇\mu

We say that a sequence of events $(A_{N})_{N\geq 1}$ is exponentially small for a sequence of probability measures $(m_{N})_{N\geq 1}$ if there are constants $\delta,c_{1},c_{2}>0$ such that for any $N$ we have

Proof. First note that ${\mathcal{H}}_{\nu}\geq{\mathcal{H}}_{\mu}$, so $Z_{\nu}\leq Z_{\mu}$. We claim that the following inequality holds:

To prove it, by Jensen’s inequality we have

Let $(A_{N})_{N\geq 1}$ be now a sequence of events exponentially small for $\mu$. By (3.23) we have

so $(A_{N})_{N\geq 1}$ is also exponentially small for $\nu$.

Assume now that $(A_{N})_{N\geq 1}$ is exponentially small for $\nu$: there are constants $\delta,c_{1},c_{2}>0$ such that for any $N$ we have

where we used $Z_{\nu}<Z_{\mu}$. Choosing $t=N^{\delta/2}$ makes the second term exponentially small, and the first one as well by using as previously Lemma 2.1, Lemma 3.1 and (3.3).

From the previously proved equivalence of the measures $\mu$ and $\nu$, we can easily obtain rigidity of the particles at scale $N^{-1/2}$,

For any $\alpha,\varepsilon>0$, there are constants $\delta,c_{1},c_{2}>0$ such that for any $N\geq 1$ and $k\in\llbracket\alpha N,(1-\alpha)N\rrbracket$,

Proof. From Lemma 3.5 about the convexity of $\mathcal{H}_{\nu}$, we get by the classical Bakry-Émery criterion [BakEme1983] that $\nu$ satisfies a logarithmic Sobolev inequality with constant of order $1/N$, so by Herbst’s lemma concentration at scale $N^{-1/2}$ holds for individual particles for $\nu$: there is a constant $c>0$ such that for any $N\geq 1$, $k\in\llbracket 1,N\rrbracket$ and $x>0$,

By Lemma 3.6, this implies that for some constants $\delta,c_{1},c_{2}>0$,

The proof will be complete if we can prove that for any $\varepsilon>0$ and $k\in\llbracket\alpha N,(1-\alpha)N\rrbracket$, for large enough $N$ we have

By Lemma 2.2, $|m_{N}-m|\to 0$ for $\eta>N^{-1/2+\varepsilon}$, because on this domain $\frac{1}{N^{2}}k_{N}\to 0$, as concentration at scale $N^{-1/2}$ holds for $\mu$. So using Lemma 2.3 we finally get that (3.26) holds, finishing the proof.

The multiscale analysis

The purpose of this paragraph is to prove the following proposition: if rigidity holds at scale $N^{-1+a}$, it holds also at scale $N^{-1+\frac{3}{4}a}$. The argument very closely follows Section 3.3 of [BouErdYau2011] and we will just explain the modifications.

Assume that for some $a\in(0,1)$ the following property holds: for any $\alpha,\varepsilon>0$, there are constants $\delta,c_{1},c_{2}>0$ such that for any $N\geq 1$ and $k\in\llbracket\alpha N,(1-\alpha)N\rrbracket$,

Then the same property holds also replacing $a$ by $3a/4$: for any $\alpha,\varepsilon>0$, there are constants $\delta,c_{1},c_{2}>0$ such that for any $N\geq 1$ and $k\in\llbracket\alpha N,(1-\alpha)N\rrbracket$, we have

Proof of Theorem 1.1. This is an immediate consequence of the initial estimate, Proposition 3.7, and iterations of Proposition 4.1.

As in Section 3.3 of [BouErdYau2011], two steps are required in the proof of the above Proposition 4.1. First we will prove that concentration holds at the smaller scale $N^{-1+\frac{a}{2}}$.

Assume that (4.1) holds. Then for any $\alpha>0$ and $\varepsilon>0$, there are constants $c_{1},c_{2},\delta>0$ such that for any $N\geq 1$ and $k\in\llbracket\alpha N,(1-\alpha)N\rrbracket$,

After the better concentration from this proposition, the rigidity can be improved to the scale $N^{-1+\frac{3a}{4}}$.

Assume that (4.1) holds. Then for any $\alpha>0$ and $\varepsilon>0$, there is a constant $c>0$ such that for any $N\geq 1$ and $k\in\llbracket\alpha N,(1-\alpha)N\rrbracket$,

where $\gamma_{k}^{(N)}$ is defined in (3.25).

Propositions 4.2 and 4.3 are the equivalent versions of Propositions 3.12 and 3.13 of [BouErdYau2011] with no convexity assumption on $V$. Proposition 4.1 can be proved exactly in the same way as Proposition 3.11 [BouErdYau2011] by using Propositions 4.2 and 4.3. Notice that this argument does not use the convexity of $V$. We now explain the proof of Propositions 4.2 and 4.3.

For the proof of Proposition 4.3, we can follow the proof of Proposition 3.13 in [BouErdYau2011] line by line. At a single place, in estimating the second term on the r.h.s. of (3.51) in [BouErdYau2011], the spectral gap inequality for $\mu$ (Eq. (3.12) in [BouErdYau2011]) was used, but the necessary estimate immediately follows from Proposition 3.7.