Abstract
We study a class of interacting particle systems on \(\mathbb {R}\) which was recently investigated by Götze and Venker (Ann Probab 42(6):2207–2242, 2014. doi:10.1214/13-AOP844). These ensembles generalize eigenvalue ensembles of Hermitian random matrices by allowing different interactions between particles. Although these ensembles are not known to be determinantal one can use the stochastic linearization method of Götze and Venker (Ann Probab 42(6):2207–2242, 2014. doi:10.1214/13-AOP844) to represent them as averages of determinantal ones. Our results describe the transition between universal behavior in the regime of the Tracy–Widom law and non-universal behavior for large deviations of the rightmost particle. Moreover, a detailed analysis of the transition that occurs in the regime of moderate deviations, is provided. We also compare our results with the corresponding ones obtained recently for determinantal ensembles (Eichelsbacher et al. in Symmetry Integr Geom Methods Appl 12:18, 2016. doi:10.3842/SIGMA.2016.093; Schüler in Moderate, large and superlarge deviations for extremal eigenvalues of unitarily invariant ensembles. Ph.D. thesis, University of Bayreuth, 2015). In particular, we discuss how the averaging effects the leading order behavior in the regime of large deviations. In the analysis of the averaging procedure we use detailed asymptotic information on the behavior of Christoffel–Darboux kernels that is uniform for perturbative families of weights. Such results have been provided by K. Schubert, K. Schüler and the authors in Kriecherbauer et al. (Markov Process Relat Fields 21(3, part 2):639–694, 2015).
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Funding was provided by Deutsche Forschungsgemeinschaft (DE) (CRC 701).
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Martin Venker has been supported by the CRC 701 at Bielefeld University.
Appendix: Asymptotics for determinantal ensembles
Appendix: Asymptotics for determinantal ensembles
In this appendix we extract those results from [24] on the asymptotics of the determinantal ensembles \(P_{N,V,f;L}:=P_{N,V-f/N;L}\), defined on \([-L,L]^N\) that are relevant in our context.
We begin by recalling the notion of the equilibrium measure with respect to a general convex external field \(\tilde{V}\). By this we mean the unique Borel probability measure \(\mu _{\tilde{V}}\) which minimizes the energy functional
A general reference on equilibrium measures with external fields is [33]. It is known that under mild assumptions such a unique minimizer exists. In our case \(\tilde{V}\) will be convex and the density of \(\mu _{\tilde{V}}\) can be described as follows (see e.g. [24, Sections 1 and 2]): There exist real numbers \(a_{\tilde{V}} < b_{\tilde{V}}\) that are uniquely determined by the two equations for a and b,
We denote by \(\lambda _{\tilde{V}}\) the linear rescaling that maps the interval \([-1, 1]\) onto \([a_{\tilde{V}}, b_{\tilde{V}}]\),
and introduce a function \(G_{\tilde{V}}\) on \(\mathbb {R}\) by
that inherits real analyticity and positivity from \(\tilde{V}''\). The density of the equilibrium measure is then given by (see [24, (2.1), (1.11–1.14)])
The linearization technique of [21] requires to consider ensembles \(P_{N,V-f/N; L}\) that arise from restricting the \(P_{N,V-f/N}\) to \([-L,L]^N\) and renormalizing them as probability measures. Note that the choice of L throughout the paper ensures that the support \([a_{V-f/N}, b_{V-f/N}]\) of \(\mu _{V-f/N}\) is contained in the interior of \([-L,L]\). Moreover, the functions f are always defined on some fixed domain D in the complex plane that contains \([-L,L]\). They belong to the real Banach space \(X_D:=\{f:\, D\, \longrightarrow \, \mathbb {C}\,:\,f\) analytic and bounded, \(f(D\cap \mathbb {R})\subset \mathbb {R}\}\), equipped with the norm \(\Vert f\Vert :=\Vert f\Vert _{D}:=\sup _{z\in D}\left|f(z)\right|\). By [24, Lemma 2.4], the maps \(\tilde{V}\mapsto a_{\tilde{V}}\), \(\tilde{V}\mapsto b_{\tilde{V}}\) that relate the external fields to the endpoints of the support of their equilibrium measures are \(C^1\) with bounded derivatives on a neighborhood of V in \(X_D\). Thus, for sufficiently large values of N,
where the subscript \(_{V,f}\) is short for \(_{V-f/N}\). We can now formulate a first result on kernels \(K_{N,V,f;L}\) associated to determinantal ensembles of the form \(P_{N,V,f;L}\) that has essentially been proved in [24].
Proposition 2
(cf. [24, Theorem 1.8]) Let \(V:\, [-L,L]\, \longrightarrow \, \mathbb {R}\) be real-analytic with \(\inf _{t\in [-L,L]}V''(t)>0\). Assume that the support \([a_V,b_V]\) of the equilibrium measure \(\mu _V\) is contained in \((-L,L)\) and recall the notations introduced in this appendix. Moreover, define
where s, \(t \in \mathbb {R}\). Then we have for fixed \(0<\kappa <1/3\), \(0< \varepsilon < \min (2/15, 2/3 -2 \kappa )\), \(q \in \mathbb {R}\),
with the \(\mathcal {O}\) terms being uniform in N, in \(f\in X_D\) with \(\Vert f\Vert _{D}\le N^\kappa \) and in s, t within the respective regions.
Proof
The result in [24, Theorem 1.8] provides leading order information on
together with error bounds that have the uniformity needed to prove Proposition 2. We note in passing that (76) ensures that, uniformly for \(f\in X_D\) with \(\Vert f\Vert _{D}\le N^\kappa \), \([a_{V,f},b_{V,f}]\subset (-L,L)\) for N large enough.
The reader should be aware of some differences in notation, e.g. the quantities Q, V, \(\gamma _V^+\), \(\mathbb {A}{\text {i}}\) in [24] correspond to V, \(V - f/N\), \(\gamma _{V,f}(b_{V,f}-a_{V,f})/2\), \(K_\text {Ai}\) in the present paper.
Our first task is to replace \(b_{V,f}\) by \(b_V\) and \(\gamma _{V,f}\) by \(\gamma _V\). Using
relations (74)–(76), the thrice differentiability of V on \([-L, L]\), the analyticity of f in D, and the positivity of \(G_V\), one obtains
uniformly for \(f\in X_D\) with \(\Vert f\Vert _{D}\le N^\kappa \), \(0<\kappa <1\) and also uniformly in \(t\in \mathbb {R}\) with both \(\lambda _V(t)\), \(\lambda _{V,f}(t) \in [-L,L]\). This yields in particular \(\gamma _{V,f}=\gamma _V(1+\mathcal {O}(\Vert f\Vert /N))\). In order to be able to apply the results of [24] we relate to any \(q \le t \le N^{\varepsilon }\) a number \(\widehat{t}=\widehat{t}(N, V, f)\) by
and in the same way we relate \(\widehat{s}\) to s. For all relevant values of f, t, and s we obtain the uniform estimates
Next we discuss the different regions in the (s, t) plane one by one.
(a) \(q \le s, t \le 2\): Application of Theorem 1.8. in [24] yields
and the claim follows from (78) and from \(\gamma _{V, f}/\gamma _V = 1 + \mathcal {O}(N^{\kappa -1})\). Observe that we need a slightly modified version of the result in [24], because one only has that \(\widehat{s}\), \(\widehat{t}\) lie in some small neighborhood of [q, 2]. However, the proof in [24] shows that the statement also holds true in such an slightly enlarged neighborhood. This remark applies to the next two cases as well.
(b) \(1 \le s, t \le N^{\varepsilon }\): Theorem 1.8 of [24] together with \(\varepsilon < 2/15\) show
The asymptotic formula for the Airy kernel (see e.g. [24, (4.23)]) implies the existence of some \(C > 1\) with
for all s, \(t \ge 1\). Using e.g. [24, Proposition 4.2] together with the asymptotics of the Airy function and its derivative (see [1, 10.4.59, 10.4.61]) one obtains a similar formula for the partial derivatives of the Airy kernel leading up to
Since \(\varepsilon < 2/3 -2\kappa \) relation (79) implies for all \(s^*\), \(t^*\) with distance \(\mathcal {O}(N^{\kappa - 1/3})\) from s, resp. t
yielding the claim.
(c) \(q \le s \le 1; 2 \le t \le N^{\varepsilon }\): This time application of [24] gives for \(\varepsilon < 2/15\)
The standard techniques used to derive the asymptotics of the Airy kernel in [24] (see in particular Proposition 4.2 there) yield the following estimates for the values of s and t considered in the present case:
where we have used \(\varepsilon < 2/3 -2\kappa \) to derive the last relation. This completes the proof.\(\square \)
In the regime of moderate and large deviations Proposition 2 does not provide the leading order description, which is the content of the following proposition. As it turns out, it suffices to consider the Christoffel–Darboux kernel on the diagonal. In order to simplify the resulting formulas we now make use of the assumed evenness of Q and h that implies the evenness of V (cf. Remark 1 b)) and hence \(a_V=-b_V\).
Proposition 3
(cf. [24, Theorem 1.5]) Let V satisfy the assumptions of Proposition 2 and assume in addition \(a_V=-b_V\). In addition to the notations introduced in this appendix, recall the definition of the function \(\eta _V\) in (15). There exist constants \(c_0 > 0\) and \(C_0 \ge 1\) only depending on V such that for
-
a)
\(b_V + C_0N^{-2/3}< t < L\):
$$\begin{aligned}&K_{N,V,f;L}(t,t)\\&\quad =\frac{b_V e^{-N\eta _V(t/b_V) + \mathcal {O}\left( \sqrt{t-b_V} \Vert f\Vert \right) }}{4\pi (t^2-b_V^2)} \left[ 1 + \mathcal {O} \left( \frac{\Vert f\Vert }{N\left|t-b_{V}\right|} \right) +\mathcal {O} \left( \frac{1}{N\left|t-b_{V}\right|^{3/2}} \right) \right] . \end{aligned}$$ -
b)
\(b_V + C_0N^{-2/3}< t < L - 1\):
$$\begin{aligned}&\int _t^L K_{N,V,f;L}(y,y) dy\\&\quad =\frac{b_V^2 e^{-N\eta _V(t/b_V) + \mathcal {O}\left( \sqrt{t-b_V} \Vert f\Vert \right) }}{4\pi N (t^2-b_V^2) \eta _V'(t/b_V)} \left[ 1 + \mathcal {O} \left( \frac{\Vert f\Vert }{N\left|t-b_{V}\right|} \right) +\mathcal {O} \left( \frac{1}{N\left|t-b_{V}\right|^{3/2}} \right) \right] . \end{aligned}$$
All \(\mathcal {O}\) terms appearing in statements a) or b) are uniform in N, in \(t\in (b_V + C_0N^{-2/3}, L)\) resp. in \(t\in (b_V + C_0N^{-2/3}, L-1)\), and in \(f\in X_D\) with \(\Vert f\Vert \equiv \Vert f\Vert _{D}\le c_0N(t-b_V)\).
Proof
From the arguments given in the beginning of the proof of Proposition 2 it follows that we can apply the result [24, Theorem 1.5 (ii)] to the ensembles \(P_{N,V-f/N; L}\) provided that \(\Vert f\Vert /N\) is sufficiently small which we can always achieve by choosing the constant \(c_0\) in the statement of Proposition 3 small enough. Hence there exists a positive number \(\tilde{C}\) such that for all \(x>1+\tilde{C}N^{-2/3}\) with \(\lambda _{V,f}(x) \le L\)
holds with the required uniformity. Setting \(x=\lambda _{V,f}^{-1}(t)\) we immediately arrive at
Using again the smallness of \(\Vert f\Vert /[N(t-b_V)]\), we learn from (76) that for real exponents \(\alpha \) one has
and consequently
We now turn to the exponential term. Applying (77) to the definition of \(\eta _V\) in (15), we see \(\eta _{V,f}=\eta _V(1+\mathcal {O}(\Vert f\Vert /N))\) uniformly on the domain of interest. Thus
Moreover, it is straightforward to see that \(\lambda _V^{-1}(t) = t/b_V\), \(\lambda _V^{-1}(t) -\lambda _{V,f}^{-1}(t) = \mathcal {O}(\Vert f\Vert /N)\), \(\eta _V (x) = \mathcal {O}(\left|x-1\right|^{3/2})\), and \(\eta _V' (x) = \mathcal {O}(\left|x-1\right|^{1/2})\). Using in addition (81) we obtain
that leads to
Combining (80) with (82) and (83) completes the proof of statement a).
The essential relation for establishing the second claim is
with an \(\mathcal {O}\) term that is uniform in N, t, and f in their respective domains and where \(a = a_{V,f}\), \(b = b_{V,f}\) and \(\eta = \eta _{V,f} \circ \lambda _{V,f}^{-1}\). We first sketch the derivation of (84) following [35, Lemma 4.8] (cf. [17]). Since \(\eta \) is strictly increasing the substitution \(u(y) := \eta (y)-\eta (t)\) is invertible and we obtain
Observe that \(k(u) = k(0) + u k'(\tilde{u})\) for some \(\tilde{u} \in (0, u)\). In addition, straightforward estimates yield
We arrive at
with \(d = \eta (L) - \eta (L-1) > 0\). Evaluating k(0) provides (84).
Note that (84) implies in particular that \(\int _t^L K_{N,V,f;L}(y,y) dy\) has exactly the same representation as \(K_{N,V,f;L}(t,t)\) in (80) except for additional factors N and \((\eta _{V,f} \circ \lambda _{V,f}^{-1})'(t)\) in the denominator. Thus the arguments in the proof of statement a) below (80) together with
prove statement b).\(\square \)
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Kriecherbauer, T., Venker, M. Edge statistics for a class of repulsive particle systems. Probab. Theory Relat. Fields 170, 617–655 (2018). https://doi.org/10.1007/s00440-017-0765-1
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DOI: https://doi.org/10.1007/s00440-017-0765-1
Keywords
- Universality
- Airy kernel
- Tracy–Widom distribution
- Random matrices
- Moderate deviations
- Large deviations
Mathematics Subject Classification
- 60B20
- 82C22