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Sharp transition of the invertibility of the adjacency matrices of sparse random graphs

Abstract

We consider three models of sparse random graphs: undirected and directed Erdős–Rényi graphs and random bipartite graph with two equal parts. For such graphs, we show that if the edge connectivity probability p satisfies \(np\ge \log n+k(n)\) with \(k(n)\rightarrow \infty \) as \(n\rightarrow \infty \), then the adjacency matrix is invertible with probability approaching one (n is the number of vertices in the two former cases and the same for each part in the latter case). For \(np\le \log n-k(n)\) these matrices are invertible with probability approaching zero, as \(n\rightarrow \infty \). In the intermediate region, when \(np=\log n+k(n)\), for a bounded sequence \(k(n)\in \mathbb {R}\), the event \(\Omega _0\) that the adjacency matrix has a zero row or a column and its complement both have a non-vanishing probability. For such choices of p our results show that conditioned on the event \(\Omega _0^c\) the matrices are again invertible with probability tending to one. This shows that the primary reason for the non-invertibility of such matrices is the existence of a zero row or a column. We further derive a bound on the (modified) condition number of these matrices on \(\Omega _0^c\), with a large probability, establishing von Neumann’s prediction about the condition number up to a factor of \(n^{o(1)}\).

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Notes

  1. When \(\ell =1\) by a slight abuse of notation we take \(\hat{z}_1=x_{[1: \sqrt{np}]}\).

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Acknowledgements

We thank the anonymous referees for their suggestions that led to an improvement of the presentation of this paper. AB acknowledges support of the Department of Atomic Energy, Government of India (GOI), under project no. RTI4001. Research of AB was partially supported by Grant 147/15 from the Israel Science Foundation, a funding from the European Research Council under the European Unions Horizon 2020 research and innovation program (Grant Agreement Number 692452), an Infosys–ICTS Excellence Grant, and a Start-up Research Grant (SRG/2019/001376) and a MATRICS Grant (MTR/2019/001105) from Science and Engineering Research Board of GOI. Research of AB is carried out in part as a member of the Infosys-Chandrasekharan virtual center for Random Geometry, supported by a grant from the Infosys Foundation. Part of this research was performed while MR visited Weizmann Institute of Science in Rehovot, Israel, where he held Rosy and Max Varon Professorship. He is grateful to Weizmann Institute for its hospitality and for creating an excellent work environment. The research of MR was supported in part by the NSF Grant DMS 1464514 and by a fellowship from the Simons Foundation.

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Appendices

Appendix A: Structural properties of the adjacency matrices of sparse graphs

In this section we prove that certain structural properties of \(A_n\), as listed in Lemma 3.7, hold with high probability when \(A_n\) satisfies Assumption 3.1 with p such that \(np \ge \log (1/\bar{C} p)\), for some \(\bar{C} \ge 1\). We also show that under the same assumption we have bounds on the number of light columns of \(A_n\), namely we prove Lemma 3.15.

First let us provide the proof of Lemma 3.15.

Proof of Lemma 3.15

The proof is a simple application of Chernoff bound and Markov’s inequality.

Since the entries of \(A_n\) satisfies Assumption 3.1, using Stirling’s approximation we note that

$$\begin{aligned} \mathbb {P}({{\,\mathrm{col}\,}}_j(A_n) \text { is light})&\le \sum _{\ell =0}^{\delta _0 np }\left( {\begin{array}{c}n-1\\ \ell \end{array}}\right) p^{\ell } (1-p)^{n-1-\ell } \nonumber \\&\le 2\delta _0 np \left( \frac{e}{\delta _0}\right) ^{\delta _0 np} \cdot \exp (-p(n-\delta _0 np)) \nonumber \\&\le \exp \left( -np \left[ 1-\delta _0 p - \delta _0 \log \left( \frac{2e}{\delta _0} \right) \right] \right) , \end{aligned}$$
(A.1)

where in the second inequality we have used the fact that \(p \le 1/4\). Therefore, for \(np \ge C \log n\), with C large, using the union bound we find \(\mathbb {E}[ |\mathcal {L}(A_n)| ]<1/n\). Hence by Markov’s inequality we deduce that

$$\begin{aligned} \mathbb {P}(\mathcal {L}(A_n) \ne \varnothing ) = \mathbb {P}(|\mathcal {L}(A_n)| \ge 1) \le \mathbb {E}[|\mathcal {L}(A_n)|] \le 1/n. \end{aligned}$$

To prove the upper bound on the cardinality of \(\mathcal {L}(A_n)\) we note that the assumption \(np \ge \log (1/\bar{C} p)\) implies that \(np \ge (1-\delta ) \log n\), for any \(\delta >0\), for all large n. Therefore, using (A.1) and Markov’s inequality, setting \(\delta =\frac{1}{9}\), we find that for \(np \le 2 \log n\),

$$\begin{aligned} \mathbb {P}(|\mathcal {L}(A_n)| \ge n^{\frac{1}{3}}) \le n^{-{\frac{1}{3}}} \mathbb {E}|\mathcal {L}(A_n)| \le n^{\frac{2}{3}} \cdot n^{-\frac{8}{9}} \cdot n^{2 \delta _0 p +2 \delta _0 \log \left( \frac{2e}{\delta _0} \right) } \le n^{-\frac{1}{9}}, \end{aligned}$$

for all large n, whenever \(\delta _0\) is chosen sufficiently small. For p such that \(2 \log n \le np \le C_{3.15} \log n\) we note from (A.1) that

$$\begin{aligned} \mathbb {P}({{\,\mathrm{col}\,}}_j(A_n) \text { is light}) \le \frac{1}{n}, \qquad j \in [n]. \end{aligned}$$

Therefore, an union bound followed by Markov’s inequality yield the desired result. \(\square \)

Proof of Lemma 3.7

We will show that each of the six properties of the event \(\Omega _{3.7}\) hold with probability at least \(1 - Cn^{-2\bar{c}_{3.7}}\), for some constant \(C >0\). Then, taking a union bound the desired conclusion would follow.

First let us start with the proof of (3.7). Since the inequality \(np \ge \log (1/\bar{C} p)\) implies that \(np \ge \log n/2\), for all large n, it follows from Chernoff bound that property (3.7) of the event \(\Omega _{3.7}\) holds with probability at least \(1-1/n\), for all large n. We omit the details.

Next let us prove that property (3.7) of \(\Omega _{3.7}\) holds with high probability. For \((i,j) \in \left( {\begin{array}{c}[n]\\ 2\end{array}}\right) \) and \(k \in [n]\) denote by \(\Omega _{(i,j),k}\) the event that the columns \({{\,\mathrm{col}\,}}_i(A_n), {{\,\mathrm{col}\,}}_j(A_n)\) are light and \(a_{k,i}, a_{k,j} \ne 0\). Note that the event that two light columns intersect is contained in the event \(\cup _{i,j,k} \Omega _{(i,j),k}\). Therefore, we need to find bounds \(\mathbb {P}(\Omega _{(i,j),k})\). Since the entry \(a_{i,j}\) may depend on \(a_{j,i}\) we need to consider the cases \(k \in [n]\backslash \{i,j\}\) and \(k \in \{i,j\}\) separately.

First let us fix \(k \in [n]\backslash \{i,j\}\). We note that

$$\begin{aligned} \Omega _{(i,j),k} \subset \left\{ a_{k,i}=a_{k,j} =1, \, |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_i(A_n))\backslash \{i,j\}|, |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))\backslash \{i,j\}| \le \delta _0 np\right\} . \end{aligned}$$

Therefore, recalling that under Assumption 3.1 the entries of the sub-matrix of \(A_n\) indexed by \(([n]\backslash \{i,j\}) \times \{i,j\}\) are i.i.d. \({{\,\mathrm{Ber}\,}}(p)\) we obtain that

$$\begin{aligned} \mathbb {P}(\Omega _{(i,j), k}) \le p^2 \exp \left( -2np \left[ 1-\delta _0 p - \delta _0 \log \left( \frac{2e}{\delta _0} \right) \right] \right) =:q, \end{aligned}$$

for all large n, where we have proceeded similarly as in (A.1) to bound the probability of the event

$$\begin{aligned} \left\{ |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_i(A_n))\backslash \{i,j\}|, |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))\backslash \{i,j\}| \le \delta _0 np \right\} . \end{aligned}$$

Since \(np \ge \log (1/\bar{C} p)\) an application of the union bound shows that

$$\begin{aligned}&\mathbb {P}\left( \bigcup _{i \ne j \in [n], k \notin \{i,j\}}\Omega _{(i,j),k} \right) \le n \cdot \left( {\begin{array}{c}n\\ 2\end{array}}\right) q \nonumber \\&\quad \le \frac{p^{-1}}{2} e^{-np} \cdot (np)^3 \cdot \exp \left( -np \left[ 1-2\delta _0 p - 2\delta _0 \log \left( \frac{2e}{\delta _0} \right) \right] \right) \nonumber \\&\quad \le \frac{\bar{C}}{2} \cdot (np)^3 \cdot \exp \left( -np \left[ 1-2\delta _0 p - 2\delta _0 \log \left( \frac{2e}{\delta _0} \right) \right] \right) \le n^{-c}, \end{aligned}$$
(A.2)

for some absolute constant c and all large n, where we use that \(np \ge \log n/2\), which as already seen is a consequence of the assumption \(np \ge \log (1/\bar{C} p)\).

Next let us consider the case \(k \in \{i,j\}\). Without loss of generality, let us assume that \(k=i\). We see that

$$\begin{aligned} \Omega _{(i,j),i} \subset \left\{ a_{i,j} =1, \, |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_i(A_n))\backslash \{i,j\}|, |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))\backslash \{i,j\}| \le \delta _0 np\right\} . \end{aligned}$$

Hence proceeding same as above we deduce

$$\begin{aligned} \mathbb {P}\left( \bigcup _{i \ne j \in [n], k \in \{i,j\}}\Omega _{(i,j),k} \right)&\le 2 \cdot \left( {\begin{array}{c}n\\ 2\end{array}}\right) \cdot p \cdot \exp \left( -2np \left[ 1-\delta _0 p - \delta _0 \log \left( \frac{2e}{\delta _0} \right) \right] \right) \nonumber \\&\le p^{-1} e^{-np} \cdot (np)^2 \cdot \exp \left( -np \left[ 1-2\delta _0 p - 2\delta _0 \log \left( \frac{2e}{\delta _0} \right) \right] \right) \nonumber \\&\le n^{-c}. \end{aligned}$$
(A.3)

So combining the bounds of (A.2) and (A.3) we conclude that property (3.7) of \(\Omega _{3.7}\) holds with probability at least \(1- n^{-2 \bar{c}_{3.7}}\).

Now let us prove that (3.7) holds with high probability. We let \(j \in [n]\), \(I =(i_1 ,\ldots ,i_{r_0}) \in \left( {\begin{array}{c}[n]\backslash \{j \}\\ r_0\end{array}}\right) \), and \(k_1 ,\ldots ,k_{r_0} \in [n]\), for some absolute constant \(r_0\) to be determined during the course of the proof. Denote by \(\Omega _{j,I,(k_1 ,\ldots ,k_{r_0})}\) the event that all the columns indexed by I are light, and for any \(i_\ell \in I\), \(k_\ell \in {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_{i_\ell }(A_n)) \cap {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))\). Equipped with this notation we see that the event that there exists a column such that its support intersects with the supports of at least \(r_0\) light columns is contained in the event \(\cup _{j; I; k_\ell , \ell \in [r_0]} \Omega _{j,I, (k_1,k_2,\ldots ,k_{r_0})}\).

Since all the columns indexed by I are light, applying property (3.7) it follows that \(\{k_\ell \}_{\ell =1}^{r_0}\) are distinct. Therefore, for matrices with independent entries (3.7) follows upon bounding the probability of the events

$$\begin{aligned} \left| {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_{i_\ell }(A_n))\backslash \{k_\ell '\}_{\ell '=1}^{r_0}\right| \le \delta _0 np, \quad \ell \in [r_0] \end{aligned}$$

and

$$\begin{aligned} a_{k_\ell , j} = a_{k_\ell , i_\ell } =1, \qquad \ell \in [r_0], \end{aligned}$$

followed a union bound. Recall that under Assumption 3.1 the entry \(a_{i,j}\) may only depend on \(a_{j,i}\) for \(i,j \in [n]\). Therefore, to carry out this scheme for matrices satisfying Assumption 3.1 we additionally need to show that the support of \({{\,\mathrm{col}\,}}_j(A_n)\) is almost disjoint from the set of light columns with high probability, so that we can omit the relevant diagonal block to extract a sub-matrix with jointly independent entries.

To this end, we claim that

$$\begin{aligned} \mathbb {P}\left( \exists j \in [n]: |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n)) \cap \mathcal {L}(A_n)| \ge 3\right) \le n^{-c'}, \end{aligned}$$
(A.4)

for some \(c' >0\). To establish (A.4) we fix \(j \in [n]\) and note that

$$\begin{aligned}&\left\{ |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n)) \cap \mathcal {L}(A_n)| \ge 3, |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))| \le C_{3.7}np \right\} \\&\quad \subset \big \{\exists k \text { with } 2 \le k \le C_{3.7}np, \text { and } i_1, i_2, \ldots , i_k \in [n] \backslash \{j\} \text { distinct } \text {such that } \\&\qquad |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_{i_\ell }(A_n))\backslash \{i_1,i_2\ldots , i_k,j\}| \le \delta _0 np, \ell =1,2,\ldots ,k\big \}. \end{aligned}$$

For ease of writing, let us denote

$$\begin{aligned} q':= \exp \left( -np \left[ 1-\delta _0 p - C_{3.7}p- \delta _0 \log \left( \frac{2e}{\delta _0} \right) \right] \right) . \end{aligned}$$

By Assumption 3.1 the entries \(\{a_{i',j'}\}\) for \((i',j') \in \{i_\ell \}_{\ell =1}^k \times ([n]\backslash (\{i_\ell \}_{\ell =1}^k \cup \{j\})\) are jointly independent \({{\,\mathrm{Ber}\,}}(p)\) random variables. Therefore applying Stirling’s approximation once more, and proceeding similarly as in (A.1) we find that

$$\begin{aligned}&\mathbb {P}\left( |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n)) \cap \mathcal {L}(A_n)| \ge 3, |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))| \le C_{3.7}np \Big | {{\,\mathrm{col}\,}}_j(A_n)\right) \\&\quad \le \sum _{k=2}^{C_{3.7}np} \left( {\begin{array}{c}C_{3.7}np\\ k\end{array}}\right) q'^k \le \sum _{k \ge 2} \left( \frac{e C_{3.7}np}{k}\right) ^k \cdot q'^k \\&\qquad \le e^{-np} p^{-1}\cdot p \cdot \left( {e C_{3.7}np}\right) ^2 \\&\qquad \cdot \exp \left( - np \left[ 1-2\delta _0 p - 2 C_{3.7}p- 2 \delta _0 \log \left( \frac{2e}{\delta _0} \right) \right] \right) . \end{aligned}$$

Since by Lemma 3.15 we see that \(\mathcal {L}(A_n) =\varnothing \) with high probability when \(p \ge \frac{C_{3.15}\log n}{n}\). Without loss of generality, we therefore assume that \(p \le \frac{C_{3.15}\log n}{n}\). So, by the union bound over j, using the fact that \(np \ge \log (1/\bar{C} p)\) and property (3.7) of the event \(\Omega _{3.7}\) we have that, for all large n,

$$\begin{aligned} \mathbb {P}\left( \exists j \in [n]: |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n)) \cap \mathcal {L}(A_n)| \ge 3\right)&\le \bar{C} \cdot \left( {e C_{3.7}np}\right) ^3 \cdot \exp \left( - np (1-\delta )\right) +1/n\\&\le 2 \exp (-np(1-2\delta )), \end{aligned}$$

for some \(\delta >0\). This establishes the claim (A.4).

Equipped with (A.4) we turn to proving (3.7). Using (A.4) we see that excluding a set of probability at most \(n^{-c'}\), for any \(j,I,(k_1 ,\ldots ,k_{r_0})\) such that \(\Omega _{j,I,(k_1 ,\ldots ,k_{r_0})}\) occurs, we can find \(\ell _{1},\ldots , \ell _{{r_0-3}}\) with \(k_{\ell _s} \in [n]\backslash (\mathcal {L}(A_n) \cup \{j\}) \subset [n] \backslash (I \cup \{j\})\) for all \(s=1,2,\ldots ,r_0-3\). For such \(k_{\ell _s}\), all events \(|{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_{i_{\ell _s}}(A_n))\backslash (I \cup \{j\})| \le \delta _0 np\) and \(a_{k_{\ell _s}, j} = a_{k_{\ell _s}, i_{\ell _s}}=1\) with \(s=1,2,\ldots ,r_0-3\) are independent. Denote for brevity

$$\begin{aligned} \bar{q}:= \exp \left( -np \left[ 1-\delta _0 p - \delta _0 \log \left( \frac{2e}{\delta _0} \right) \right] \right) . \end{aligned}$$

Note that under the assumption \(np \ge \log (1/\bar{C} p)\) we have \(\bar{q} \le \exp ( - \log n/2)\) for all large n. Hence, recalling Assumption 3.1, using (A.4) and property (3.7) of \(\Omega _{3.7}\), and proceeding similarly as in (A.1) once again we see that

$$\begin{aligned}&\mathbb {P}\left( \bigcup _{\begin{array}{c} j, k_1,,\ldots ,, k_{r_0} \in [n]\\ I \in \left( {\begin{array}{c}[n]\backslash \{j\}\\ r_0\end{array}}\right) \end{array}} \Omega _{j,I, (k_1,,\ldots ,, k_{r_0})} \right) \nonumber \\ \le&\sum _{\begin{array}{c} j, k_1,,\ldots ,, k_{r_0} \in [n]\\ I \in \left( {\begin{array}{c}[n]\backslash \{j\}\\ r_0\end{array}}\right) \end{array}} \prod _{s=1}^{r_0-3}\mathbb {P}\left( |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_{i_{\ell _s}}(A_n))\backslash (I \cup \{j\})| \le \delta _0 np \right) \cdot \mathbb {P}\left( a_{k_{\ell _s}, j} = a_{k_{\ell _s}, i_{\ell _s}}=1 \right) + n^{-c_0} \nonumber \\ \le&n^{r_0+1} \left( {\begin{array}{c}n-1\\ r_0\end{array}}\right) p^{2(r_0-3)} \cdot \bar{q}^{r_0-3} +n ^{-c'} \le (np)^{2(r_0-3)} \cdot n^7 \cdot \bar{q}^{r_0-3} + n^{-c'}\le n^{-\bar{c}} + n^{-c_0}, \end{aligned}$$
(A.5)

for some \(\bar{c}, c_0 >0\), where the last step follows upon choosing \(r_0\) such that \(r_0 - 3 > 15\). This completes the proof of property (3.7).

Next let us show that (3.7) holds with high probability. First we will prove that for any \(j \in [n]\) such that \({{\,\mathrm{col}\,}}_j(A_n)\) is normal we have

$$\begin{aligned} \left| {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n)) \cap \left( \bigcup _{i \in \mathcal {L}(A_n)} {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_i(A_n)) \right) \right| \le \frac{\delta _0}{64} np, \end{aligned}$$
(A.6)

with high probability. Note that the difference between (A.6) and property (3.7) of \(\Omega _{3.7}\) is that in (A.6) it is claimed that for any \(j \in [n]\) such that \({{\,\mathrm{col}\,}}_j(A_n)\) is normal its support does not have a large intersection with that of light columns. To establish property (3.7) we need to strengthen the above to deduce that one can replace the matrix \(A_n\) by its folded version on the lhs of (A.6) with the loss of factor of four in its rhs.

Turning to prove (A.6), we see that if (3.7) holds then given any \(j \in [n]\) there exists only \(r_0\) light columns \({{\,\mathrm{col}\,}}_{i_1}(A_n), ,\ldots ,, {{\,\mathrm{col}\,}}_{i_{r_0}}(A_n)\) such that their supports intersect that of \({{\,\mathrm{col}\,}}_j(A_n)\). Hence,

(A.7)

Since by (3.7) we have that \(|{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))| \le C_{3.7} np\), using Stirling’s approximation and a union bound we show that the event on the rhs of (A.7) holds with small probability.

Indeed, for \(i \ne j \in [n]\), denoting

$$\begin{aligned} \bar{\Omega }_{i,j}: = \left\{ \left| {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n)) \cap {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_i(A_n)) \right| \ge \frac{\delta _0}{64 r_0}np\right\} , \end{aligned}$$

and using the fact that property (3.7) holds with high probability we deduce that

$$\begin{aligned} \mathbb {P}\left( \bigcup _{i \ne j \in [n]} \bar{\Omega }_{i,j} \right)&\le \sum _{i \ne j \in [n]} \mathbb {E}\left[ \mathbb {P}\left( \bar{\Omega }_{i,j} \cap \left\{ |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))| \le C_{3.7}np \right\} \Big | {{\,\mathrm{col}\,}}_j(A_n)\right) \right] + n^{-1}\nonumber \\&\le \left( {\begin{array}{c}n\\ 2\end{array}}\right) \cdot \left( {\begin{array}{c}C_{3.7}np\\ \frac{\delta _0}{64r_0} np\end{array}}\right) p^{\frac{\delta _0}{64r_0} np} + n^{-1} \nonumber \\&\le n^2 \cdot \left( \frac{e C_{3.7} 64 r_0 p}{\delta _0} \right) ^{\frac{\delta _0}{64r_0} np} + n^{-1} \le 2 n^{-1}, \end{aligned}$$
(A.8)

for all large n. Thus combining (A.7) and (A.8) and applying property (3.7) of the event \(\Omega _{3.7}\) we establish that (A.6) holds with probability at least \(1- n^{-\widetilde{c}}\) for some \(\widetilde{c} >0\).

As mentioned above, to show that property (3.7) holds with high probability we need to strengthen (A.6). To this end, recalling the definition of the folded matrix (see Definition 3.5) we note that \(k \in {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_i({{\,\mathrm{fold}\,}}(A_n)) \cap {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j({{\,\mathrm{fold}\,}}(A_n))\) implies that

$$\begin{aligned} k \in {{\,\mathrm{supp}\,}}_{\mathfrak {u}}({{\,\mathrm{col}\,}}_i(A_n)) \cap {{\,\mathrm{supp}\,}}_{\mathfrak {v}}({{\,\mathrm{col}\,}}_j(A_n)) \end{aligned}$$

for some \(\mathfrak {u}, \mathfrak {v}\in \{1,2\}\), where for any \(\ell \in [n]\).

$$\begin{aligned} {{\,\mathrm{supp}\,}}_1({{\,\mathrm{col}\,}}_\ell (A_n)) := {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_\ell (A_n)) \cap [\mathfrak {n}], \end{aligned}$$
$$\begin{aligned} {{\,\mathrm{supp}\,}}_2({{\,\mathrm{col}\,}}_\ell (A_n)) := ({{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_\ell (A_n)) \cap [\mathfrak {n}+1, 2\mathfrak {n}]) - \mathfrak {n}, \end{aligned}$$

and for any set \(S \subset [n]\) and \(k \in \mathbb {Z}\) we denote \(S+k :=\{x+k: x \in S\}\). Using the observation we see that it suffices to show that

$$\begin{aligned} \left| {{\,\mathrm{supp}\,}}_\mathfrak {u}({{\,\mathrm{col}\,}}_j(A_n)) \cap \left( \bigcup _{i \in \mathcal {L}(A_n)} {{\,\mathrm{supp}\,}}_\mathfrak {v}({{\,\mathrm{col}\,}}_i(A_n)) \right) \right| \le \frac{\delta _0}{64} np, \end{aligned}$$
(A.9)

with high probability, for all \(\mathfrak {u}, \mathfrak {v}\in \{1,2\}\). If \(\mathfrak {u}=\mathfrak {v}\) then (A.9) is an immediate consequence of (A.6). It remains to prove (A.9) for \(\mathfrak {u}\ne \mathfrak {v}\). Let us consider the case \(\mathfrak {u}=1\) and \(\mathfrak {v}=2\). From (A.4) we have

$$\begin{aligned} \mathbb {P}\left( \exists j \in [n]: |{{\,\mathrm{supp}\,}}_1({{\,\mathrm{col}\,}}_j(A_n)) \cap \mathcal {L}(A_n) | \ge 3\right) \le n^{-c'}. \end{aligned}$$

Therefore, proceeding similarly as in the steps leading to (A.5) we deduce that, with the desired high probability, for any \(j \in [n]\), such that \({{\,\mathrm{col}\,}}_j(A_n)\) is a normal column, there are at most \(r_0\) light columns \(\{{{\,\mathrm{col}\,}}_{i_\ell }(A_n)\}_{\ell =1}^{r_0}\) so that \({{\,\mathrm{supp}\,}}_1({{\,\mathrm{col}\,}}_j(A_n)) \cap {{\,\mathrm{supp}\,}}_2({{\,\mathrm{col}\,}}_{i_\ell }(A_n)) \ne \varnothing \). Now arguing similarly as in the proof of (A.6) we derive (A.9) for \(\mathfrak {u}=1\) and \(\mathfrak {v}=2\). The proof of the other case is similar and hence is omitted.

Next we show that (3.7) holds with high probability. We first fix an \(I \subset [n]\) with \(2 \le |I| \le c_{3.7} p^{-1}\) and derive that (3.7) holds with certain probability for each such choice of I and then take an union over I.

Since the entry \(a_{i,j}\) may depend on \(a_{j,i}\), for \(i \ne j\), to derive that (3.7) holds with the desired probability we need to split it into two cases. Namely, the off-diagonal and the diagonal blocks require separate arguments. First we consider the off-diagonal block.

To this end, define the random variables

$$\begin{aligned} \eta _i:=\max (|\{j\in I:\,\mathfrak {a}_{i,j}\ne 0\}|-1,0), \quad \quad i\in [\mathfrak {n}]\backslash \bar{I}, \end{aligned}$$

where we recall \(\bar{I}:=\bar{I}(I):=\{j \in [\mathfrak {n}]: j \in I \text { or } j +\mathfrak {n}\in I\} \subset [\mathfrak {n}]\), \(\mathfrak {n}:=\lfloor n/2 \rfloor \), and \(\mathfrak {a}_{i,j}\) denotes the (ij)th entry of \({{\,\mathrm{fold}\,}}(A_n)\). Observe that

$$\begin{aligned}\left| \bigcup _{j\in I}\left( {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j({{\,\mathrm{fold}\,}}(A_n)))\backslash \bar{I}\right) \right| = \sum _{j\in I}\left| {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j({{\,\mathrm{fold}\,}}(A_n)))\backslash \bar{I}\right| -\sum _{i \in [\mathfrak {n}]\backslash \bar{I}} \eta _i. \end{aligned}$$

To prove (3.7) we need to show that \(\sum \eta _i\) cannot be too large with large probability. To show the latter we use the standard Laplace transform method.

Note that

$$\begin{aligned} \mathfrak {a}_{i,j} = \xi _{i,j} \cdot \delta _{i,j}, \quad i \in [\mathfrak {n}]\backslash \bar{I}, j \in I, \end{aligned}$$

where \(\{\xi _{i,j}\}\) are i.i.d. Rademacher random variables, \(\delta _{i,j}\) are i.i.d. \({{\,\mathrm{Ber}\,}}(\mathfrak {p})\) random variables, and \(\mathfrak {p}:=2p(1-p)\). Therefore,

$$\begin{aligned} \mathbb {P}\{\eta _i=\ell \}\le {|I|\atopwithdelims ()\ell +1}\mathfrak {p}^{\ell +1}, \quad \ell \in \mathbb {N}. \end{aligned}$$

Thus, for any \(\lambda >0\) such that \(e^\lambda \mathfrak {p}|I|\le 1\), we have

$$\begin{aligned} \mathbb {E}\left( e^{\lambda \eta _i}\right) \le 1+\sum _{\ell =1}^\infty \big (e^\lambda \big )^{\ell }\,\mathfrak {p}^{\ell +1}|I|^{\ell +1}((\ell +1)!)^{-1}\le 1+e \mathfrak {p}|I|, \end{aligned}$$

and hence

$$\begin{aligned} \mathbb {P}\left\{ \sum _{i \in [\mathfrak {n}]\backslash \bar{I}} \eta _i \ge t\right\} \le \frac{\big (1+e\mathfrak {p}|I|\big )^{|[\mathfrak {n}]\backslash \bar{I}|}}{\exp (\lambda t)},\quad \quad t>0. \end{aligned}$$

In particular, taking \(t:=\frac{\delta _0}{32} n p|I|\) and \(\lambda :=\log \frac{1}{\mathfrak {p}|I|}\), we get

$$\begin{aligned} \mathbb {P}\left\{ \sum _{i\in [\mathfrak {n}]\backslash \bar{I}} \eta _i\ge \frac{\delta _0}{32} np |I|\right\}&\le \exp \left( e\mathfrak {p}\mathfrak {n}|I|-\lambda \frac{\delta _0}{32} np |I|\right) \le \exp \left( -\lambda \frac{\delta _0}{64} np |I|\right) \nonumber \\&\le \exp \left( - \log \left( \frac{1}{2 c_{3.7}}\right) \cdot \frac{\delta _0}{64}np |I|\right) \le n^{-2|I|}, \end{aligned}$$
(A.10)

where the second and the third inequalities follow from recalling that \( p |I| \le c_{3.7}\) for some sufficiently small constant \(c_{3.7}\), depending only on \(\delta _0\), and the last inequality follows from our assumption that \(np \ge \log n/2\) and shrinking \(c_{3.7}\) even further, if necessary.

To complete the proof of the fact that (3.7) holds with high probability, we show that

$$\begin{aligned} \mathbb {P}\left( \sum _{j\in I}\left| {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j({{\,\mathrm{fold}\,}}(A_n)))\cap \bar{I}\right| \ge \frac{\delta _0}{32} np |I| \right) \le 2 n^{-2|I|}. \end{aligned}$$
(A.11)

Now the proof finishes from (A.10) and (A.11) by first taking a union over \(I \in \left( {\begin{array}{c}[n]\\ k\end{array}}\right) \) followed by a union over \(k=2,3,\ldots , c_{3.7} p^{-1}\). We omit the details.

Turning to prove (A.11), we denote \( \hat{I}(I):=\hat{I}:= \cup _{i \in \bar{I}} \{ i , \mathfrak {n}+i\} \). As the entries of \(A_n\) are \(\{0,1\}\)-valued, we see that

$$\begin{aligned} {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j({{\,\mathrm{fold}\,}}(A_n))) \cap \bar{I} \subset {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n)) \cap \hat{I}. \end{aligned}$$

Moreover, \(I \subset \hat{I}\). Therefore, it is enough to show that

$$\begin{aligned} \mathbb {P}\left( \sum _{j\in \hat{I}}\left| {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))\cap \hat{I}\right| \ge \frac{\delta _0}{32} np |I|\right) \le 2 n^{-2|I|}. \end{aligned}$$
(A.12)

Since \(A_n\) satisfies Assumption 3.1 we have that the upper triangular part of the sub-matrix of \(A_n\) induced by the rows and columns indexed by \(\hat{I}\) consists of independent \(\{0,1\}\)-valued random variables stochastically dominated by i.i.d. \({{\,\mathrm{Ber}\,}}(p)\) variables. So does the lower triangular part of that sub-matrix.

For ease of writing let us write

$$\begin{aligned} \mathscr {X}_L:= \sum _{i \ge j \in \hat{I}} a_{i,j} \quad \text { and } \quad \mathscr {X}_U:=\sum _{i \le j \in \hat{I}} a_{i,j} \end{aligned}$$

and note \(\mathscr {X}_U\) and \(\mathscr {X}_L\) has the same law. Thus to establish (A.12) it suffices to show that

$$\begin{aligned} \mathbb {P}(\mathscr {X}_U \ge \frac{\delta _0}{64} np |I|) \le n^{-2|I|}. \end{aligned}$$
(A.13)

The above is obtained by using the Laplace transform method as above. Indeed, we note that

$$\begin{aligned} \mathbb {P}(\mathscr {X}_U = \ell ) \le \left( {\begin{array}{c}|\hat{I}|^2\\ \ell \end{array}}\right) p^\ell , \quad \ell \in \mathbb {N}\cup \{0\} \end{aligned}$$

and therefore

$$\begin{aligned} \mathbb {E}\left[ \exp \left( \lambda X_U\right) \right] \le \exp \left( e^\lambda p |\hat{I}|^2 \right) \le \exp (4 |I|), \end{aligned}$$

where \(\lambda = \log \frac{1}{p |I|}\) and we have used the fact that \(|\hat{I}| \le 2 |I|\). Hence, upon using Markov’s inequality and proceeding similarly as in (A.10) we deduce (A.13). It completes the proof of (A.12).

Now it remains to prove that property (3.7) holds with high probability. Recalling the definition of the folded matrix again we note that \(|{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j({{\,\mathrm{fold}\,}}(A_n)))| \le |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))|\). To show that the cardinality of the support of \({{\,\mathrm{col}\,}}_j({{\,\mathrm{fold}\,}}(A_n))\) is not too small compared to its unfolded version we observe that if \(k \in {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))\) but \(k \notin {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j({{\,\mathrm{fold}\,}}(A_n)))\) then we must have that \(a_{k,j}=a_{k,\mathfrak {n}+j}=1\). Using estimates on the binomial probability and Chernoff bound we show that number of such k is small.

To carry out the above heuristic, we fix \(j \in [n]\) and since the entries of \(A_n\) are \(\{0,1\}\) valued we note that

$$\begin{aligned} |{{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j({{\,\mathrm{fold}\,}}(A_n))) | = \sum _{i \in [\mathfrak {n}]} \left[ a_{i,j} \cdot (1-a_{i+\mathfrak {n}, j}) + a_{i+\mathfrak {n},j} \cdot (1-a_{i, j})\right] \end{aligned}$$

Further observe that

$$\begin{aligned} \left| {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))\cap [\mathfrak {n}]\right| = \sum _{i=1}^{\mathfrak {n}} a_{i,j} = \sum _{i=1}^{\mathfrak {n}} a_{i,j} \cdot a_{i+\mathfrak {n}, j} + \sum _{i=1}^\mathfrak {n}a_{i,j} \cdot (1-a_{i+\mathfrak {n}, j}) \end{aligned}$$

and

$$\begin{aligned} \left| {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))\cap ([2\mathfrak {n}]\backslash [\mathfrak {n}])\right| = \sum _{i=\mathfrak {n}+1}^{2\mathfrak {n}} a_{i,j} = \sum _{i=1}^{\mathfrak {n}} a_{i,j} \cdot a_{i+\mathfrak {n}, j} + \sum _{i=1}^\mathfrak {n}a_{i+\mathfrak {n},j} \cdot (1-a_{i, j}). \end{aligned}$$

Therefore,

$$\begin{aligned} \left| \left| {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j(A_n))\right| - \left| {{\,\mathrm{supp}\,}}({{\,\mathrm{col}\,}}_j({{\,\mathrm{fold}\,}}(A_n))) \right| \right| \le 2 \sum _{i=1}^{\mathfrak {n}} a_{i,j} \cdot a_{i+\mathfrak {n}, j} +1. \end{aligned}$$

Denoting

$$\begin{aligned} \Delta _j:= \sum _{i=1}^{\mathfrak {n}} a_{i,j} \cdot a_{i+\mathfrak {n}, j}, \end{aligned}$$

we see that \(\Delta _j\) is stochastically dominated by \({{\,\mathrm{Bin}\,}}(\mathfrak {n}, p^2)\). To finish the proof we need to find bounds on \(\Delta _j\).

First let us consider the case \(p \le n^{-5/12}\). For any \(k_0 \in \mathbb {N}\), sufficiently large, we see that

$$\begin{aligned} \mathbb {P}(\Delta _j \ge k_0) \le \left( {\begin{array}{c}\mathfrak {n}\\ k_0\end{array}}\right) p^{2k_0} \le (np^2)^{k_0} \le n^{-k_0/6} \le n^{-2}. \end{aligned}$$
(A.14)

For \( n^{-5/12} \le p \le c\), for some small \(c >0\) depending on \(\delta _0\), we use Chernoff bound to deduce that

$$\begin{aligned} \mathbb {P}\left( \Delta _j \ge \frac{\delta _0}{16}np\right) \le \mathbb {P}\left( \Delta _j \ge 2 p^{-1/2} \cdot \mathfrak {n}p^2 \right) \le \exp \left( -\frac{1}{3} p^{-1/2} \cdot \mathfrak {n}p^2 \right) \le \exp \left( -\frac{1}{9} n^{3/8} \right) .\nonumber \\ \end{aligned}$$
(A.15)

Combining (A.14) and (A.15) and taking an union over \(j \in [n]\) we show that property (3.7) holds with high probability. This completes the proof of the lemma. \(\square \)

Appendix B: Proof of invertibility over sparse vectors with a large spread component

In this section we prove Proposition 3.21. As already mentioned in Sect. 3.3 the proof is similar to that of Proposition 3.18. There are two key differences. Since our goal is to find a uniform bound on \(\Vert A_n x \Vert _2\) for x’s with a large spread component, unlike in the proof of Proposition 3.21, we use Lemma 3.22 to estimate the small ball probability. Moreover, as noted earlier, Assumption 3.1 allows some dependencies among its entries. Therefore, to tensorize the small ball probability we need to extract a sub-matrix of \(A_n\) with jointly independent entries such that the coordinates of x corresponding to the columns of this chosen sub-matrix form a vector with a large spread component and a sufficiently large norm. Below we make this idea precise.

Proof of Proposition 3.21

First, let us show that (3.41) implies (3.43). To this end, we begin by noting that if \(c_{3.21} < \frac{1}{2}\) then for any \(x \in \mathrm{Dom}(c_0^*n, c_{3.21}K^{-1})\) we have that \(\Vert x_{[M_0+1: c_0^*n]}\Vert _2 \ge \Vert x_{[c_0^*n+1:n]}\Vert _2\) (see also (3.34)). Hence, for \(x \notin V_{M_0}\) we obtain that \(\Vert x_{[M_0+1: c_0^*n]}\Vert _2 \ge \rho /\sqrt{2}\). Therefore, (3.41) implies that

$$\begin{aligned} \mathbb {P}\left( \left\{ \exists x \in V_{c_0^*,{c}_{3.21}}\backslash V_{M_0}: \left\| (A_n-p {\varvec{J}}_n ) x -y \right\| _2 \le 2\widetilde{c}_{3.21}\rho \sqrt{np}\right\} \cap \Omega _K^0\right) \le \exp (-2\bar{c}_{3.21}n), \end{aligned}$$

where we recall the definition of \(\Omega _K^0\) from (3.29). Hence, proceeding as in the steps leading to (3.31) we deduce (3.43) upon assuming (3.41).

So, to complete the proof of the proposition it remains to establish (3.41). To prove it, we fix \(x \notin V_{M_0}\). Then

$$\begin{aligned} \Vert x_{[M_0+1:n]}\Vert _2 \ge \rho \quad \text { and } \quad \frac{\Vert x_{[M_0+1:n]}\Vert _\infty }{\Vert x_{[M_0+1:n]}\Vert _2} \le \frac{K}{c_{3.18}}\cdot \sqrt{\frac{\log n}{n \sqrt{\log \log n}}}. \end{aligned}$$

Fixing \(\bar{c}_0 \in (c_0^*,1)\), as \(M_0 \le \frac{1-\bar{c}_0}{2}n\) for all large n, recalling the fact that the non-zero entries of \(x_{[m_1: m_2]}\), for \(m_1 <m_2\), are the coordinates of x that take places from \(m_1\) to \(m_2\) in the non-increasing arrangement according their absolute values, we note that

$$\begin{aligned} \Vert x_{[M_0+1:n]}\Vert _2^2 = \Vert x_{[M_0+1:(1-\bar{c}_0)n]}\Vert _2^2 + \Vert x_{[(1-\bar{c}_0)n+1:n]}\Vert _2^2 \le \frac{1+\bar{c}_0}{1- \bar{c}_0} \cdot \Vert x_{[M_0+1:(1-\bar{c}_0)n]}\Vert _2^2.\nonumber \\ \end{aligned}$$
(B.1)

Therefore

$$\begin{aligned}&\Vert x_{[M_0+1:(1-\bar{c}_0)n]}\Vert _2 \ge \rho \cdot \sqrt{\frac{1-\bar{c}_0}{1+\bar{c}_0}} \quad \text { and } \quad \frac{\Vert x_{[M_0+1:(1-\bar{c}_0)n]}\Vert _\infty }{\Vert x_{[M_0+1:(1-\bar{c}_0)n]}\Vert _2} \\&\quad \le \frac{K}{c_{3.18}} \cdot \sqrt{\frac{1+\bar{c}_0}{1-\bar{c}_0}}\cdot \sqrt{\frac{\log n}{n\sqrt{\log \log n}}}. \end{aligned}$$

Note that this shows \(x_{[M_0+1:(1-\bar{c}_0)n]}\) has a large spread part and a large norm. Denoting \(\mathcal {I}:=\mathcal {I}(x):= {{\,\mathrm{supp}\,}}(x_{[M_0+1:(1-\bar{c}_0)n]})\) we note that Assumption 3.1 implies that the entries \(\{a_{i,j}\}_{j \in \mathcal {I}, i \notin \mathcal {I}}\) are i.i.d. \({{\,\mathrm{Ber}\,}}(p)\). So, now we can carry out the scheme that was outlined above by using the joint independence of \(\{a_{i,j}\}_{j \in \mathcal {I}, i \notin \mathcal {I}}\).

Indeed, using Lemma 3.22 we find that for any \(i \notin \mathcal {I}\), \(y \in \mathbb {R}^n\), and \(\varepsilon _0>0\) we have

$$\begin{aligned}&\mathbb {P}\left( \left| ((A_n - p {\varvec{J}}_n) x)_i -y_i\right| \le p^{1/2} (1-p)^{1/2} \Vert x_{[M_0+1:(1-\bar{c}_0)n]}\Vert _2 \cdot \bar{c}_0^{-1/2}\varepsilon _0 \right) \nonumber \\&\quad \le \mathcal {L}\left( \langle {\varvec{a}}_i, x\rangle , p^{1/2} (1-p)^{1/2} \Vert x_{[M_0+1:(1-\bar{c}_0)n]}\Vert _2 \cdot \bar{c}_0^{-1/2}\varepsilon _0 \right) \nonumber \\&\quad \le C_{3.22} \left( \frac{\varepsilon _0}{\sqrt{\bar{c}_0}} + \frac{2K}{c_{3.18}} \cdot \sqrt{\frac{1+\bar{c}_0}{1-\bar{c}_0}}\cdot \sqrt{\frac{\log n}{np\sqrt{\log \log n}}} \right) \le 2 C_{3.22} \frac{\varepsilon _0}{\sqrt{\bar{c}_0}}, \end{aligned}$$
(B.2)

for all sufficiently large n (depending only on \(\varepsilon _0\)), where \({\varvec{a}}_i\) is the ith row of \(A_n\) and we have used the fact that \(np \ge c_1 \log n\) for some \(c_1 >0\). We will choose \(\varepsilon _0\) as a small constant during the course of the proof.

Since the entries \(\{a_{i,j}\}_{j \in \mathcal {I}, i \notin \mathcal {I}}\) are i.i.d. \({{\,\mathrm{Ber}\,}}(p)\), we apply a standard tensorization argument, for example [42, Lemma 5.4], to deduce from (B.2) that for any \(x \notin V_{M_0}\)

$$\begin{aligned}&\mathbb {P}\left( \Vert (A_n - p {\varvec{J}}_n) x - y\Vert _2 \le \sqrt{np(1-p)} \Vert x_{[M_0+1:(1-\bar{c}_0)n]}\Vert _2 \varepsilon _0\right) \nonumber \\&\le \mathbb {P}\left( \sum _{i \notin \mathcal {I}}\left| ((A_n - p {\varvec{J}}_n) x)_i -y_i\right| ^2 \le p(1-p) \Vert x_{[M_0+1:(1-\bar{c}_0)n]}\Vert _2^2 \varepsilon _0^2 \bar{c}_0^{-1}\cdot |\mathcal {I}^c|\right) \le \left( C_0 \cdot \varepsilon _0\right) ^{\bar{c}_0 n},\nonumber \\ \end{aligned}$$
(B.3)

for some constant \(C_0\), depending only on \(\bar{c}_0\), where the last two steps follow from the fact that \(|\mathcal {I}^c| \ge \bar{c}_0 n\) and upon choosing \(\varepsilon _0\) such that \( C_0 \cdot \varepsilon _0 \le \frac{1}{2}\).

To complete the proof we use an \(\varepsilon \)-net similar to the proof of Proposition 3.18. First, setting

$$\begin{aligned} \varepsilon =\frac{\rho }{2}\tau = \frac{\varepsilon _0 \rho }{448 K} \cdot \sqrt{\frac{1-\bar{c}_0}{1+\bar{c}_0}}, \end{aligned}$$
(B.4)

and using Fact 3.20 we obtain a net \(\widetilde{\mathcal {M}}\) in \(V_{c_0^*,{c}_{3.21}} \backslash V_{M_0}\) with

$$\begin{aligned} |\widetilde{\mathcal {M}}|\le & {} \bar{C}^n {n \atopwithdelims ()M_0}{n \atopwithdelims ()c_0^* n} \left( \frac{1}{\varepsilon _0}\right) ^{c_0^*n+1} \left( \frac{1}{\rho }\right) ^{M_0+1}\\\le & {} \bar{C}^n \left( \frac{en}{M_0}\right) ^{M_0} \left( \frac{e}{c_0^*}\right) ^{c_0^* n} \left( \frac{1}{\varepsilon _0}\right) ^{c_0^*n+1} \left( \frac{1}{\rho }\right) ^{M_0+1}, \end{aligned}$$

for some \(\bar{C}\), depending only on \(\bar{c}_0\) and \(c_0^*\). Recalling that \(M_0 = \frac{n \sqrt{\log \log n}}{\log n}\) and the definition of \(\rho \) we observe that

$$\begin{aligned} \left( \frac{n}{M_0\rho ^2}\right) ^{M_0} = \exp (o(n)), \end{aligned}$$

for \(p \in (0,1/2]\) satisfying \(np \ge c_1 \log n\). Therefore, we further have that

$$\begin{aligned} |\widetilde{\mathcal {M}}| \le C_\star ^n \cdot \left( \frac{1}{\varepsilon _0}\right) ^{c_0^*n+1}, \end{aligned}$$
(B.5)

for some other constant \(C_\star \), depending only on \(c_0^*\) and \(\bar{c}_0\). Next proceeding as in the steps leading to (3.37) we obtain that for any \(x \in V_{c_0^*,{c}_{3.21}} \backslash V_{M_0}\) there exists \(\bar{x} \in \widetilde{\mathcal {M}}\) such that for any \(y \in \mathbb {R}^n\)

$$\begin{aligned}&\Vert (A_n - p {\varvec{J}}_n) \bar{x} - y\Vert _2 \le \Vert (A_n- p {\varvec{J}}_n) x - y\Vert _2 + 4K \sqrt{np}\cdot \varepsilon \\&\quad + 2K \sqrt{np} \cdot \tau \cdot \Vert v_{\bar{x}}\Vert _2 + 12 c_{3.21} \sqrt{np} \cdot \left\| v_{\bar{x}} \right\| _2. \end{aligned}$$

Since \(\Vert \bar{x}_{[M_0+1:c_0^*n]}\Vert _2 = \Vert v_{\bar{x}}\Vert _2 \ge \rho /\sqrt{2}\), using (B.4) and setting

$$\begin{aligned} c_{3.21} \le \frac{\varepsilon _0}{56} \cdot \sqrt{\frac{1-\bar{c}_0}{1+\bar{c}_0}}, \end{aligned}$$
(B.6)

we deduce from above that any \(x \in V_{c_0^*,{c}_{3.21}} \backslash V_{M_0}\) there exists \(\bar{x} \in \widetilde{\mathcal {M}}\) such that for any \(y \in \mathbb {R}^n\)

$$\begin{aligned} \Vert (A_n - p {\varvec{J}}_n) \bar{x} - y\Vert _2 \le \Vert (A_n- p {\varvec{J}}_n) x - y\Vert _2 +\frac{\varepsilon _0}{7} \cdot \sqrt{\frac{1-\bar{c}_0}{1+\bar{c}_0}} \Vert \bar{x}_{[M_0+1:c_0^*n]}\Vert _2 \cdot \sqrt{np}. \end{aligned}$$

Furthermore, by our construction of the net \(\widetilde{\mathcal {M}}\),

$$\begin{aligned} \Vert {x}_{[M_0+1:c_0^*n]}\Vert _2 \le \Vert \bar{x}_{[M_0+1:c_0^*n]}\Vert _2+ \varepsilon \le \left( 1 + \frac{\varepsilon _0}{224 K}\right) \cdot \Vert \bar{x}_{[M_0+1:c_0^*n]}\Vert _2. \end{aligned}$$

Therefore, upon assuming \(p \le \frac{1}{4}\) and recalling (B.1), this further yields that

$$\begin{aligned}&\mathbb {P}\left( \exists {x} \in V_{c_0^*,{c}_{3.21}} \backslash V_{M_0}: \Vert (A_n- p {\varvec{J}}_n) {x} -y \Vert _2 \le \frac{\varepsilon _0}{4}\Vert {x}_{[M_0+1:c_0^*n]}\Vert _2 \cdot \sqrt{\frac{1-\bar{c}_0}{1+\bar{c}_0}}\cdot \sqrt{np} \right) \nonumber \\&\quad \le \mathbb {P}\left( \exists \bar{x} \in \widetilde{\mathcal {M}}: \Vert (A_n- p{\varvec{J}}_n) \bar{x} -y \Vert _2 \le \sqrt{np(1-p)} \Vert \bar{x}_{[M_0+1:(1-\bar{c}_0)n]}\Vert _2 \varepsilon _0 \right) \nonumber \\&\quad \le |\widetilde{\mathcal {M}}| \cdot \left( C_0 \cdot \varepsilon _0\right) ^{\bar{c}_0 n} \le C_0^{\bar{c}_0 n} C_\star ^n \varepsilon _0^{-1}\varepsilon _0^{(\bar{c}_0 - c_0^*)n} \le \varepsilon _0^{\frac{\bar{c}_0- c_0^*}{2} n}, \end{aligned}$$
(B.7)

where the second last step follows from (B.5) and the last step follows upon using the fact that \(\bar{c}_0 > c_0^*\) and choosing \(\varepsilon _0\) sufficiently small. This yields (3.41) and hence the proof of the proposition is complete. \(\square \)

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Basak, A., Rudelson, M. Sharp transition of the invertibility of the adjacency matrices of sparse random graphs. Probab. Theory Relat. Fields 180, 233–308 (2021). https://doi.org/10.1007/s00440-021-01038-4

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  • DOI: https://doi.org/10.1007/s00440-021-01038-4

Keywords

  • Random matrices
  • Sparse matrices
  • Erdős–Rényi graph
  • Invertibility
  • Smallest singular value
  • Condition number

Mathematics Subject Classification

  • 46B09
  • 60B20