Abstract
We consider asymptotics of the correlation functions of characteristic polynomials corresponding to random weighted \(G(n, \frac{p}{n})\) Erdős–Rényi graphs with Gaussian weights in the case of finite p and also when \(p \rightarrow \infty \). It is shown that for finite p the second correlation function demonstrates a kind of transition: when \(p < 2\) it factorizes in the limit \(n \rightarrow \infty \), while for \(p > 2\) there appears an interval \((-\lambda _*(p), \lambda _*(p))\) such that for \(\lambda _0 \in (-\lambda _*(p), \lambda _*(p))\) the second correlation function behaves like that for Gaussian unitary ensemble (GUE), while for \(\lambda _0\) outside the interval the second correlation function is still factorized. For \(p \rightarrow \infty \) there is also a threshold in the behavior of the second correlation function near \(\lambda _0 = \pm 2\): for \(p \ll n^{2/3}\) the second correlation function factorizes, whereas for \(p \gg n^{2/3}\) it behaves like that for GUE. For any rate of \(p \rightarrow \infty \) the asymptotics of correlation functions of any even order for \(\lambda _0 \in (-2, 2)\) coincide with that for GUE.
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Acknowledgments
The author is grateful to Prof. M. Shcherbina for statement of the problem and for fruitful discussions. The author is thankful to the Akhiezer Foundation for partial financial support. The author also thanks reviewers for carefully reading of the paper and for useful comments.
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Appendix
Appendix
1.1 Grassmann Variables
Consider the set of formal variables \(\{\psi _j,\, \overline{\psi }_j\}_{j=1}^n\) which satisfy the anticommutation relations
This set generates a graded algebra \(\mathcal {A}\), which is called the Grassmann algebra. Taking into account that \(\psi _j^2 = \overline{\psi }_j^2 = 0\), we have that all elements of \(\mathcal {A}\) are polynomials of \(\{\psi _j,\, \overline{\psi }_j\}_{j=1}^n\). We can also define functions of Grassmann variables. Let \(\chi \) be an element of \(\mathcal {A}\) and f be any analytical function. By \(f(\chi )\) we mean the element of \(\mathcal {A}\) obtained by substituting \(\chi - z_0\) in the Taylor series of f near \(z_0\), where \(z_0\) is a free term of \(\chi \). Since \(\chi - z_0\) is a polynomial of \(\{\psi _j,\, \overline{\psi }_j\}_{j=1}^n\) with zero free term, there exists \(l \in \mathbb {N}\) such that \((\chi - z_0)^l = 0\), and hence the series terminates after a finite number of terms.
The integral over the Grassmann variables is a linear functional, defined on the basis by the relations
A multiple integral is defined to be the repeated integral. Moreover “differentials” \(\{d\psi _j,\, d\overline{\psi }_j\}_{j=1}^n\) anticommute with each other and with \(\{\psi _j,\, \overline{\psi }_j\}_{j=1}^n\). Hence for a function f
we have by definition
The use of Grassmann variables for computing averages of determinants rests on the following identity, valid for any \(n \times n\) matrix A:
The one more identity is the Hubbard-Stratonovich transformation
which valid for any complex numbers y, t and any positive number a. The identities (6.2) also hold when y, t are arbitrary even Grassmann variables (i.e. sums of the products of even number of Grassmann variables). For even Grassmann variables the formulas can be proved by Taylor-expanding \(e^{2axy}\) and \(e^{ay(u + iv) + at(u - iv)}\) into the series and integrating each term.
The properties explained so far suffice to obtain the integral representation for \(F_{2m}\), \(m = 1\), whereas the general case \(m > 1\) requires some additional preliminaries, pertaining to antisymmetric tensor products. Further details about antisymmetric tensor products may be found in [30, Chapter 8.4].
1.1.1 Grassmann Variables and the Exterior Product
The exterior product of vectors is well-known, as well as the exterior product of alternating multilinear forms (see [30]). However, to prove Proposition 1 we need the exterior product of alternating operators. Define it as following. Let A be a linear operator on \(\Lambda ^q \mathbb {C}^n\) and B be a linear operator on \(\Lambda ^r \mathbb {C}^n\). Then the exterior product \(A \wedge B\) is the restriction of the linear operator \({\text {Alt}} \circ (A \otimes B)\) on the \(\Lambda ^{q + r} \mathbb {C}^n\). Here \({\text {Alt}}\) is the operator of the alternation, i.e.,
where \(S_k\) is the group of permutations of length k; \({\text {sgn}} \pi \) is the sign of permutation \(\pi \); \(f_\pi \) is the canonical automorphism of \(V^{\otimes k}\), which carries \(v_1 \otimes \ldots \otimes v_k\) to \(v_{\pi (1)} \otimes \ldots \otimes v_{\pi (k)}\), \(v_j \in V\); V is some finite-dimensional linear space. Note, that for \(A \in {{\mathrm{End}}}V\) the exterior product \(A \wedge A\) coincides with the well-known second exterior power of linear operator A.
Fix some basis \(\{ e_j \}_{j = 1}^n\) of \(\mathbb {C}^n\). Let \(A \in {{\mathrm{End}}}\Lambda ^k \mathbb {C}^n\) and \(\alpha , \beta \in I_{n,k}\), where \(I_{n, k}\) is defined in (2.2). By \(A_{\alpha \beta }\) we denote the corresponding entry of the matrix of A in the basis \(\{e_{\alpha _1} \wedge \ldots \wedge e_{\alpha _k},\, \alpha \in I_{n,k}\}\).
To obtain the integral representation for \(F_{2m}\) we use the lemma:
Lemma 6
Let A and B be linear operators on \(\Lambda ^q \mathbb {C}^n\) and \(\Lambda ^r \mathbb {C}^n\) respectively. Then
Proof
Let \(S_{q, r}\) be the set of such \(\pi \in S_{q + r}\) that satisfy inequalities \(\pi (1) < \ldots < \pi (q)\) and \(\pi (q + 1) < \ldots < \pi (q + r)\). Then
where
On the other hand,
where \(e_\alpha = e_{\alpha _1} \wedge \ldots \wedge e_{\alpha _q}\), \(\alpha \in I_{n, q}\).
Hence,
Thus,
which completes the proof of the lemma. \(\square \)
We also need some properties of the exterior product of the operators.
Proposition 2
Let \(A_j \in {\text {End}} \Lambda ^{q_j} \mathbb {C}^n\), \(j = \overline{1, k}\), and \(B \in {\text {End}} \mathbb {C}^n\). Then
-
(i)
\(A_1 \wedge A_2 = A_2 \wedge A_1\);
-
(ii)
\((A_1 \wedge A_2) \wedge A_3 = A_1 \wedge (A_2 \wedge A_3)\);
-
(iii)
\(\bigwedge \limits _{j = 1}^k A_j = \Big ( {\text {Alt}} \circ \bigotimes \limits _{j = 1}^k A_j \Big )\Big |_{\Lambda ^q \mathbb {C}^n}\);
-
(iv)
\(\bigwedge \limits _{j = 1}^k A_jB^{\wedge q_j} = \Big ( \bigwedge \limits _{j = 1}^k A_j \Big )B^{\wedge q}\) and \(\bigwedge \limits _{j = 1}^k B^{\wedge q_j}A_j = B^{\wedge q}\Big ( \bigwedge \limits _{j = 1}^k A_j \Big )\);
where \(q = \sum \limits _{j = 1}^k q_j\), \(B^{\wedge q} = \underbrace{B \wedge \ldots \wedge B}_{q \text { times}}\).
Proof
Assertions (i) and (ii) follow from Lemma 1 and from Grassmann variables multiplication’s anticommutativity and associativity.
(iii) Consider the case \(k = 3\).
where I is the identity operator.
The general case follows from the induction.
(iv) By definition, we have
Consider \(({\text {Alt}} \circ B^{\otimes q_j})(v_1 \otimes \ldots \otimes v_{q_j})\), \(v_l \in \mathbb {C}^n\)
Therefore, \({\text {Alt}} \circ B^{\otimes q_j} = B^{\otimes q_j} \circ {\text {Alt}}\), in particular, \(B^{\wedge q_j} = \left. B^{\otimes q_j} \right| _{\Lambda ^{q_j} \mathbb {C}^n}\). Thus,
The proof of the second formula is similar. \(\square \)
1.2 The Harish-Chandra/Itsykson–Zuber formula
For computing the integral over the unitary group, the following Harish-Chandra/Itsykson–Zuber formula is used
Proposition 3
Let A be a normal \(n \times n\) matrix with distinct eigenvalues \(\{a_j\}_{j = 1}^n\) and \(B = {{\mathrm{diag}}}\{b_j\}_{j = 1}^n\), \(b_j \in \mathbb {R}\). Then
where z is come constant, \(\Delta (A) = \Delta (\{a_j\}_{j = 1}^n)\), \(\Delta \) is defined in (2.1). Moreover, for any symmetric domain \(\Omega \) and any symmetric function f(B) of \(\{b_j\}_{j = 1}^n\)
where \(dB = \prod \nolimits _{j = 1}^{n} db_j\).
For the proof see, e.g., [16, Appendix 5].
Remark 2
Notice, that (6.3) is also valid if A has equal eigenvalues.
Indeed, if \(a_{j_1} = a_{j_2}\), \(j_1 \ne j_2\), the integrand in r.h.s. of (6.3) changes sign when \(b_{j_1}\) and \(b_{j_2}\) are swapped. Thus, the ratio of the integral and \(a_{j_1} - a_{j_2}\) is well defined.
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Afanasiev, I. On the Correlation Functions of the Characteristic Polynomials of the Sparse Hermitian Random Matrices. J Stat Phys 163, 324–356 (2016). https://doi.org/10.1007/s10955-016-1486-z
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DOI: https://doi.org/10.1007/s10955-016-1486-z