Convergence in High Probability of the Quantum Diffusion in a Random Band Matrix Model
Abstract
We consider Hermitian random band matrices H in \(d \geqslant 1 \) dimensions. The matrix elements \(H_{xy},\) indexed by \(x, y \in \varLambda \subset \mathbb {Z}^d,\) are independent, uniformly distributed random variable if \(|x-y| \) is less than the band width W, and zero otherwise. We update the previous results of the converge of quantum diffusion in a random band matrix model from convergence of the expectation to convergence in high probability. The result is uniformly in the size \(|\varLambda | \) of the matrix.
Keywords
Quantum dynamics Random matrix theory Anderson model1 Introduction
Random band matrices \(H=\left( H_{xy}\right) _{x,y \in \varGamma }\) represent systems on a large finite graph with a metric. They are the natural intermediate models to study quantum propagation in disordered systems as they interpolate in between the Wigner matrices and Random Schrödinger operators. The elements \(H_{xy}\) are independent random variables with variance \(\sigma _{xy}^2=\mathbb {E}|H_{xy}|^2\) depending on the distance between the two sites. The variance decays with the distance on the scale W, called the band width of the matrix H. This terminology comes from the simplest model in which the graph is a path on N vertices labelled by \(\varGamma =\{1,2,\ldots , N\},\) and the matrix elements \(H_{xy}\) are zero if \(|x-y| \geqslant W.\) If \(W=O(1)\) we obtain the one-dimensional Anderson type model (see [1]) and if \(W=N\) we recover the Wigner matrix. In the general Anderson model, introduced in [1], a random on-site potential V is added to a deterministic Laplacian on a graph that is typically a regular box in \(\mathbb {Z}^d\,.\) For higher dimensional models in which the graph is \(\varGamma \) is a box in \(\mathbb {Z}^d\), see [2].
In [3] it was proved that the quantum dynamics of d-dimensional band matrix is given by a superposition of heat kernels up to time scales \(t \ll W^{d/3}\,.\) Note that diffusion is expected to hold for \(t \sim W^{2}\) for \(d=1\) and up to any time for \(d \geqslant 3\) when the thermodynamic limit is taken. The threshold d / 3 on the exponent is due to technical estimates on Feynman graphs.
The approach of this paper is similar with the one in [3]. We normalize the entries of the matrix such that the rate of quantum jumps is of order one. In contrast with [3] in this paper double-rooted Feynman graphs are used to estimate the variance of the quantum diffusion. The main result of this paper is upgrading the previous results on the convergence of expectation of the quantum diffusion from [3] to convergence in high probability.
2 Model and Main Result
Throughout our investigation we will use the simplified notation \(\sum \limits _{y_1}\) for\(\sum \limits _{y_1 \in \varLambda _N}\).
Our main result gives an estimate for the variance of the random variable \(Y_T(\phi )\) up to time scales \(t=O(W^{d\kappa })\) if \(\kappa <1/3\) .
Theorem 1
Remark 1
Using the estimate that we obtain in Theorem 2.1 and Chebyshev inequality for the second moment we obtain the convergence in high probability of the random variable \(Y_{T}(\phi )\,.\) We think that the same technique can be implemented for a graphical representation with 2p directed chains with \(p \in \mathbb {N}\,.\) This approach should give similar estimates on the 2p-th moment of our random variable that we further use in the Chebyshev’s inequality to get the desired conclusion.
3 Graphical Representation
In this section we give the exact formula of the quantity of our analysis and we motivate the graphical representation that we will use in order to compute the upper bound.
3.1 Expansion in Non-backtracking Powers
Lemma 1
3.2 Graphical Representation
The predecessor vertex a(e) and the successor vertex b(e) of the edge e
The graphical representation of the rooted directed chains
Using that \(\mathbb {E}H_{xy}=0\) , it is not hard to see that the graphical representation of the variance yields to the following result (for further details, see [4]) .
Lemma 2
Lemma 3
3.3 Collapsing of Parallel Bridges
The following result is to check from the definition of \(\mathfrak {G}\); see Lemma 7.4 (ii) in [3] .
Lemma 4
Let \(\{e, e'\} \in \varSigma \). Then e and \(e'\) are adjacent only if \(e\cap e' \in V_w(\varSigma )\,.\)
Graphical representation of the skeleton for a given configuration
In the following we rewrite the right hand side of (2.7) using the summation over skeleton pairings \(\varSigma =S(\varPi )\), followed by different ways of expanding the bridges of \(\varSigma \) . For this, let \(\varPi =G_{l_\varSigma }(\varSigma )\) . We further define \(|l_{\varSigma }|:=\sum _{\sigma \in \varSigma }l_{\sigma }\) for \(\varSigma \in \mathfrak {G}\) and \(l_{\varSigma } \in \mathbb {N}^{\varSigma }\). For the skeleton \(\varSigma \in \mathfrak {G}\) of the pairing \(\varPi =G_{l_{\varSigma }}(\varSigma )\) we use the notation \(n_{ij}(\varSigma , l_{\varSigma })\) for \(n_{ij}(\varPi )\), for all \(i, j \in \{1,2\}\) .
Lemma 5
The following result follows easy from the definition of \(S_{xy}\).
Lemma 6
- (i)
\(\sum \limits _{y}(S^l)_{xy}=(\frac{M}{M-1})^{l}\,.\)
- (ii)
\((S^l)_{xy}\;\leqslant \;(\frac{M}{M-1})^{l-1}\frac{1}{M-1}\,.\)
3.4 Orbits of Vertices
Let us fix \(\varSigma \in \mathfrak {G}\) . On the set of vertices \( V(\varSigma )\) we construct the \(\textit{orbits of vertices}\) as in [3] . We define \(\tau : V(\varSigma ) \rightarrow V(\varSigma )\) as follows. Let \(i \in V(\varSigma )\) and let e be the unique edge such that \(\{\{i, b(i)\}, e\} \in \varSigma \) . Then, for any vertex i of \(\varSigma \in \mathfrak {G}\) we define \(\tau i :=b(e)\). We denote the orbit of the vertex \(i \in \varSigma \) by \([i]\;:=\; \{ \tau ^n i : n \in \mathbb {N}\}\) .
We order the edges of \(\varSigma \) in some arbitrary fashion and denote this order by < . Each bridge \(\sigma \in \varSigma \) ”sits between” the orbits \(\zeta _1(\sigma )\) and \(\zeta _2(\sigma )\). More precisely, let \(\sigma =\{e , e'\}\) with \( e <e'\) and \(e=\{ i, b(i)\}\) . Then, \(\zeta _1(\sigma ):=[i]\) and \(\zeta _2(\sigma ):=[b(i)]\) .
The construction of the orbit [i] for the vertex i
The following result is an adaptation of the \(\textit{2/3-rule}\) introduced in Lemma 7.7 of [3] .
Lemma 7
Proof
Let \(Z'(\varSigma ):=Z(\varSigma )\setminus \{ [r(\mathcal {L}_1)], [r(\mathcal {L}_2)], [s(\mathcal {L}_1)], [s(\mathcal {L}_2)]\}\) . Using the same reasoning as in the proof of the \(\textit{2/3 rule}\) in [3] we obtain that each orbit contains at least 3 vertices.
The total number of vertices of \(\varSigma \) not including \(\{r(\mathcal {L}_1),r(\mathcal {L}_2),s(\mathcal {L}_1),s(\mathcal {L}_2) \} \) is \(2|\varSigma |-4\) . It follows that \(3|Z'(\varSigma )|\;\leqslant \; 2|\varSigma |-4 \Leftrightarrow |Z'(\varSigma )|\;\leqslant \; 2|\varSigma |/3-4/3\).
Using that \(|Z^*(\varSigma )|\; \leqslant \; |Z'(\varSigma )|+2\), we obtain \(|Z^*(\varSigma )|\;\leqslant \; 2|\varSigma |/3+2/3\) . \(\square \)
We remark that Lemma 2.7 is sharp in the sense that there exists \(\varSigma \in \mathfrak {G}\) such that the estimate of Lemma 2.7 saturates.
4 The Case \(|\varSigma |\;\geqslant \; 3\)
Using Lemma 2.7 and the same argument as in Section 7.5 of [3] we obtain the following result.
Lemma 8
4.1 Estimation of the Variance for \(|l_{\varSigma }| \ll M^{1/3}\)
Lemma 9
- (i)
For any time t and for any \(n \in \mathbb {N}\) we have \(|a_n(t)|\leqslant \frac{Ct^n}{n!}\,, \) for some constant \(C\,.\)
- (ii)
We have \(\sum \limits _{n\geqslant 0}|a_n(t)|^2=1+O(M^{-1})\,,\) uniformly in \(t \in \mathbb {R}\) .
A new estimate on \( \sum _{l_{\varSigma }}\varvec{\mathrm {1}}(|l_{\varSigma }|\;\leqslant \; M^{\mu })|a_{n_{11}(\varSigma , l_{\varSigma })}(t)\overline{a_{n_{12}(\varSigma , l_{\varSigma })}(t)}a_{n_{21}(\varSigma , l_{\varSigma })}(t)\overline{a_{n_{22}(\varSigma , l_{\varSigma })}(t)}|\) is established in the following lemma. The new technique is based on splitting the summation according to the bridges that touch the rooted directed chains (Fig. 5).
Lemma 10
The directed paths \(\mathcal {S}_1, \mathcal {S}_2, \mathcal {S}_3\) and \(\mathcal {S}_4\)
Proof
Let \(\varSigma \in \mathfrak {G}\) . We denote each path by \(\mathcal {S}_1 \equiv r(\mathcal {L}_1(\varSigma ))\rightarrow s(\mathcal {L}_1(\varSigma ))\), \(\mathcal {S}_2 \equiv s(\mathcal {L}_1(\varSigma ))\rightarrow r(\mathcal {L}_1(\varSigma ))\), \(\mathcal {S}_3 \equiv r(\mathcal {L}_2(\varSigma ))\rightarrow s(\mathcal {L}_2(\varSigma ))\) and \(\mathcal {S}_4 \equiv s(\mathcal {L}_2(\varSigma ))\rightarrow r(\mathcal {L}_2(\varSigma ))\) . There always exists a bridge connecting \(\mathcal {S}_{i}\) and \(\mathcal {S}_{j}\) , for \(i \ne j\,.\) Without loss of generality we choose \(\sigma _1\) connecting \(\mathcal {S}_1\) and \(\mathcal {S}_2\) .
We have the following cases :
(i) There is a bridge \(\sigma _2 \in \varSigma \) between \(\mathcal {S}_3\) and \(\mathcal {S}_4\) .
Let \(\bar{\varSigma }\; :=\; \varSigma \setminus \{\sigma _1, \sigma _3 \}\) . There exist the functions \(f_1(l_{\bar{\varSigma }})\), \(f_2(l_{\bar{\varSigma }})\), \(f_3(l_{\bar{\varSigma }})\) and \(f_4(l_{\bar{\varSigma }})\) such that \(n_{11}(\varSigma , l_{\varSigma })=f_1(l_{\bar{\varSigma }})+l_{\sigma _1}\) and \( n_{12}(\varSigma , l_{\varSigma })=f_2(l_{\bar{\varSigma }})+l_{\sigma _1}\), \(n_{21}(\varSigma , l_{\varSigma })=f_3(l_{\bar{\varSigma }})+l_{\sigma _2}\) and \( n_{22}(\varSigma , l_{\varSigma })=f_4(l_{\bar{\varSigma }})+l_{\sigma _2}\) . Note that \(n_{11}(\varSigma , l_{\varSigma })\) and \(n_{21}(\varSigma , l_{\varSigma })\) do not represent the same linear combination of elements of \(l_\varSigma \) .
Now the claim follows like in (i). \(\square \)
4.2 Estimation of the Variance for \(|l_{\varSigma }| \geqslant M^{1/3}\)
5 Estimation for the Variance in the Case \(|\varSigma |\;\leqslant \;2\)
5.1 Estimation for the Variance in the Case \(|\varSigma |\;=\;0\) and \(|\varSigma |\;=\;1\)
5.2 Estimation of the Variance in the Case \(|\varSigma |\;=\;2\)
Given that the two rooted directed chains are connected we obtain that the graph with \(l_{\sigma _1}\) bridges that touch \(\mathcal {S}_1\) and \(\mathcal {S}_2\) and \(l_{\sigma _2}\) bridges that touch \(\mathcal {S}_3\) and \(\mathcal {S}_4\) gives no contribution to the value of the variance. Also, we obtain, up to permutations, four different possible configurations. In all four cases it holds that \(y_1=y_2\) .
Notes
Acknowledgements
This result is based on a Semester Project in ETH Zürich under the supervision of Prof. Dr. Antti Knowles. The author is grateful to Prof. Antti Knowles for the careful guiding into understanding the problem.
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