arXiv:1801.05577v2 [math.PR] 18 Jul 2018 lxne .Lta naLtv osatnTikhomirov Konstantin Lytova Anna Litvak E. Alexander h ako admrglrdgah fconstant of digraphs regular random of The hwta h akof rank the that show Keywords: 60C05 46B09, secondary: Classification: 2010 AMS prtosfrmti-etrmlilcto) nti pap model. certain this effi a more In from computationally matrices c square multiplication). are in random -vector matrices interest sparse for general that operations of known are is matrices it complexit sparse understanding invertible for Well important are quantitativ number particular, subject In singular a applications. is and matrices results square many discrete random of Singularity Introduction 1 ity. lsl eae oterno oe rmtepeiu paragr previous the from model random the to related closely admdirected random u n od[] ewudas iet eto eae ok o works related [ work mention Nguyen’s to and like 22] 19, also [7, would matrices We Bernoulli symmetric [4]. 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(large) a be ioeTmzkJeemn ireYoussef Pierre Tomczak-Jaegermann Nicole admrglrgah,rno arcs ak singularity rank, matrices, random graphs, regular random G ( ,p n, d / rglrgahon graph -regular admvralswt h parameter the with variables random 1 hc sasmercrandom symmetric a is which ) A n sa least at is rmr:6B0 55,4B6 05C80; 46B06, 15B52, 60B20, primary: degree n Abstract n n − etcs ihteuiomdsrbto.We distribution. uniform the with vertices, ≥ ihpoaiiygigt n as one to going probability with 1 1 2 d let , 1 / √ n + 2 A × n n n × o eteajcnymti fa of matrix adjacency the be arxaeiid Bernoulli i.i.d. are matrix (1) n siae ntesmallest the on estimates e s r of ors sdb olo n[1 12], Koml´os [11, by in ssed arxwoeoff-diagonal whose matrix  nigcnetr asserts conjecture anding n rw elwt sparse with deal we er 1 ihaln itr and history long a with p 0,weerno 0 random where 20], bandb Bourgain, by obtained / fsm algorithms. some of y h case The . mue cec,and science, omputer ietdErd˝os–Renyidirected econsidered. re 2+ p.I 8 Costello [8] In aph. A wthnsada and switchings e in rqieless (require cient o n (1) rcso random of trices . iglrt of singularity n  n Currently, . p probabil- 1 = n grows / is 2 ± / 1 1 and Vu proved that, given c > 1, with large probability the rank of the of G(n,p) is equal to the number of non-isolated vertices whenever c ln n/n ≤ p 1/2. It is known that p = ln n/n is the threshold of connectivity, so that when ≤ c > 1 and c ln n/n p 1/2, the graph G(n,p) typically contains no isolated ≤ ≤ vertices and is therefore of full rank with probability going to one as n tends to infinity (see [1] for quantitative bounds in the non-symmetric setting). It was also shown that if p 0 and np , then rk G(n,p) /n 1 as n goes to infinity, → → ∞  → where rk(A) stands for the rank of the matrix A. The case p = y/n for a fixed y was studied in [3] where asymptotics for rk G(n,p) /n were established.  In the absence of independence between the matrix entries, the problem of sin- gularity involves additional difficulties. Such a problem was considered for the (symmetric) adjacency matrix Mn of a random (with respect to the uniform prob- ability) undirected d- on n vertices, i.e., a graph in which each vertex has precisely d neighbours. The case d = 1 corresponds to a which is non-singular, and for d = 2 the graph is a union of cycles and the matrix is almost surely singular. Moreover, the invertibility of the adjacency matrix of the complementary graph is equivalent to that of the original one (in fact, the ranks of the adjacency matrices of a d-regular graph and of its complementary graph are the same). This can be seen by first noticing that the eigenvalues of Jn Mn, where − Jn is the n n matrix of ones, are equal to the difference between those of Jn and × those of Mn (since the two commute) and that all eigenvalues of Mn are bounded in absolute value by d, which is smaller than the only non-zero eigenvalue of Jn (equals to n). In parallel to the Erd˝os–Renyi model, Costello and Vu raised the following problem: “For what d is the adjacency matrix Mn of full rank almost surely?” (see [8, Section 10]). They conjectured that for every 3 d n 3, the ≤ ≤ − adjacency matrix Mn is non-singular with probability going to 1 as n tends to . ∞ This conjecture was mentioned again in the survey [23, Problem 8.4] and 2014 ICM talks by Frieze [9, Problem 7] and by Vu [24, Conjecture 5.8]. In the present paper, we are interested in behaviour of adjacency matrices of random directed d-regular graphs with the uniform model, that is, random graphs uniformly distributed on the set of all directed d-regular graphs on n vertices. By a directed d-regular graph on n vertices we mean a graph such that each vertex has precisely d in-neighbours and d out-neighbours and where loops and 2-cycles are allowed but multiple edges are prohibited. The adjacency matrix An of such a graph is uniformly distributed on the set of all (not necessarily symmetric) 0/1 matrices with d ones in every row and every column. As in the symmetric case, in the case d = 1 the matrix A1 is a permutation matrix which is non-singular, and in the case d = 2 the matrix A2 is almost surely singular. It is natural to ask the same question as in [8] for directed d-regular graphs (see, in particular, [5, Conjecture 1.5]). Cook [5] proved that such a matrix is asymptotically almost 2 2 surely non-singular for ω(ln n) d n ω(ln n), where f = f(n) = ω(an) ≤ ≤ − means f/an as n . Further, in [13, 14], the authors of the present → ∞ → ∞ 3 paper showed that the singularity probability is bounded above by C ln d/√d for C d n/ ln2 n, where C is a (large) absolute positive constant. This settles the ≤ ≤ problem of singularity for d = d(n) growing to infinity with n at any rate. Moreover, quantitative bounds on the smallest singular value for this model were derived in [6] and [15]. Those estimates turn out to be essential in the study of the limiting spectral distribution [6, 17].

2 The challenging case when d is a constant remains unresolved and is the main motivation for writing this note. The lack of results in this setting constitutes a major obstacle in establishing the conjectured non-symmetric (oriented) Kesten– McKay law as the limit of the spectral distribution for the directed random d-regular graph (see, in particular, [2, Section 7]). This note illustrates a partial progress in this direction. Our main result is the following theorem. Note that the probability bound in it is non-trivial only if ln n>C ln2 d, however in the complementary case we have rk (An)= n with high probability as was mentioned above. Theorem 1.1. There exists a universal constant C > 0 such that for any integer d C the following holds. Let n > d and let An be the adjacency matrix of the ≥ random directed d-regular graph on n vertices, with uniform distribution allowing loops but no multiple edges. Then 2 P rk An n 1 1 C ln d/ ln n. { ≥ − }≥ − This theorem is “one step away” from proving the conjectured invertibility for a (large) constant d. We would like to emphasize that the main point of the theorem is that even for a constant d the probability of a “good” event tends to 1 with n (and not with d as in [13, 14]). To the best of our knowledge, it is the first result of such a kind dealing with singularity of d-regular random matrices. The proof of Theorem 1.1 uses the standard technique of simple switchings (in particular, it was also used in [5] and [14]). We recall the procedure using the matrix language. Denote by n,d the set of all adjacency matrices of directed d-regular graphs on M n vertices, i.e., all 0/1 matrices with d ones in every row and every column. Given A = (ast)1≤s,t≤n n,d we say that a switching in (i,j,k,ℓ) can be performed if ∈ M aik = ajℓ = 1 and aiℓ = ajk = 0. Further, given such a matrix A n,d, we ∈ M say that a matrix A¯ = (¯ast)s,t n,d is obtained from A by a simple switching ∈ M (in (i,j,k,ℓ)) ifa ¯ik =a ¯jℓ = 0,a ¯iℓ =a ¯jk = 1, anda ¯st = ast otherwise. Note that this operation does not destroy the d-regularity of the underlying graph. A well known application of the simple switching is due to McKay [18] in the context of undirected d-regular graphs. Starting from a matrix A n,d, one can reach any ∈ M other matrix in n,d by iteratively applying simple switchings. In this connection, M a feasible strategy in estimating the cardinality of a subset n,d is to pick an B ⊂ M element in and bound the number of switchings which would result in another B element of versus switchings leading outside of . In a sense, one studies the B B stability of under this operation. We will make this standard approach more B precise in the preliminaries. Clearly, for a matrix A n,d and any 1 i < j n, ∈ M ≤ ≤ rk A = dim span (Rs)s6=i,j,Ri + Rj,Ri , { } where R1,R2, ..., Rn denote the rows of A. Since (Rs)s6=i,j and Ri + Rj are invariant under any switching involving the i-th and j-th rows, then for any matrix A¯ obtained from A by such a switching, we have

rk A rk A¯ 1. (1) | − |≤ In a sense, we will show that given a matrix from n,d of corank at least 2, most M of simple switchings tend to increase the rank. We will use that the kernel of A, ⊥ ker A, is contained in F , where Fij := span (Rℓ)ℓ6=i,j,Ri + Rj . Note that Fij is ij { } invariant under any simple switching on the i-th and j-th rows.

3 In this paper, the simple switching procedure is combined with a recent delocal- ization result for eigenvectors of An established by the authors in [16, Corollary 1.2]. Below we state a less general version of the delocalization result. Theorem 1.2 ([16]). There exists a universal constant C > 0 such that for any integer d C the following holds. Let n > d and let An be the adjacency matrix ≥ of the directed random d-regular graph on n vertices. Then with probability at least T 1 2/n any vector x ker An ker A 0 satisfies − ∈ ∪ n  \ { } 2 λ R i n : xi = λ Cn ln d/ ln n. ∀ ∈ |{ ≤ }| ≤ In fact in [16] the assumption d exp(c√ln n) was also involved, however for ≤ d exp(c√ln n) the bound on the cardinality trivially holds. A more general ≥ quantitative version of Theorem 1.2, proved in [16], served as the key element in establishing the circular law [17] for the limiting spectral distribution when the degree d = d(n) ln96 n tends to infinity with n (for the regime d> ln96 n see [6]). ≤ In this note, we take advantage of the fact that the results of [16] also work for any large constant d.

2 Preliminaries

For an n n matrix A, we denote by (Rs)s≤n and (Cols)s≤n its rows and columns re- × spectively. Given positive integer m, we denote by [m] the set 1, 2, ..., m . Further, n { } for a vector x R , we denote its support by supp x = i n : xi = 0 . ∈ ′ ′ { ≤ 6 } Given two sets , and a Q , we set Q( )= b∈B Q(b) and −1 ′ B−1B ′ ⊂B×B B S Q ( )= ′ ′ Q (b ), where B Sb ∈B Q(b)= b′ ′ : (b, b′) Q and Q−1(b′)= b : (b, b′) Q , { ∈B ∈ } { ∈B ∈ } for any b and any b′ ′. In what follows, we consider the symmetric relation ∈B ∈B Q0 on n,d n,d defined by M × M (A, A¯) Q0 if and only if A¯ can be obtained from A by a simple switching. (2) ∈ The following simple claim will be used to compare cardinalities of two sets given a relation on their Cartesian product. We refer to [14, Claim 2.1] for a proof of a similar claim. Claim 2.1. Let Q be a finite relation on ′ such that for every b and every ′ ′ −1 ′ B×B ∈B b one has Q(b) sb and Q (b ) tb′ for some numbers sb,tb′ 0. Then ∈B | |≥ | |≤ ≥

sb tb′ . X ≤ X b∈B b′∈B′

Next, given A n,d, we estimate the number of possible switchings on A, that ∈ M is, the cardinality of the set

A = (i,j,k,ℓ) : a switching in (i,j,k,ℓ) can be performed . F { } Recall that we say that a simple switching can be performed in (i,j,k,ℓ) if aik = ajℓ = 1 and aiℓ = ajk = 0. Note that this automatically implies that i = j and 6 k = ℓ. Note also that two formally distinct simple switchings (i,j,k,ℓ) and (j, i, ℓ, k) 6 result in the same transformation of a matrix.

4 Lemma 2.2. Let 1 d n and A n,d. Then ≤ ≤ ∈ M 2 2 2 n(n d)d nd(d 1) A n(n d)d . − − − ≤ |F |≤ −

Proof. To find a possible switching, we first fix an entry aik equal to 1. By d- regularity of A, there are exactly nd choices of the pair (i, k). To be able to perform a simple switching in (i,j,k,ℓ), the pair of indices (j, ℓ) must satisfy

ajℓ = 1 and (j, ℓ) / T := [n] supp Ri suppColk [n] . ∈  ×  [  × 

By d-regularity we observe that the number p of pairs (s,t) T with ast = 1 satisfies 2 2 2 ∈ d p d + (d 1) . Since there are nd choices for (j, ℓ) with ajℓ = 1, we observe ≤ ≤ − that the number q of pairs (s,t) / T with ast = 1 satisfies ∈ (n d)d (d 1)2 q (n d)d. − − − ≤ ≤ −

Since A = ndq, we obtain the desired result. |F | Remark 2.3. Note that for d = 1, i.e., in the case of a permutation matrix, the upper and lower bounds in the above lemma coincide and both equal n(n 1). More − generally, assume n = md for an integer m and consider the block- A with m d d blocks, each block consisting of ones. Then a switching in (i,j,k,ℓ) can × 2 be performed if and only if i, j correspond to different blocks (there are m(m 1)d = 2 − n(n d) such pairs) and k suppRi, ℓ suppRj (there are d such choices). Thus − ∈ ∈ for such a matrix A we have

2 A = n(n d)d , |F | − which corresponds to the upper bound in Lemma 2.2.

Denote by the event in Theorem 1.2. As usual, we don’t distinguish between E1.2 events for the uniformly distributed on n,d and corresponding 2 M subsets of n,d. In particular, denoting βn := Cn ln d/ ln n, M T := A n,d : x ker A ker A 0 λ R i n : xi = λ βn , E1.2 { ∈ M ∀ ∈ ∪ \{ } ∀ ∈ |{ ≤ }| ≤ 2 where βn := min(n,Cn ln d/ ln n) and C is the constant from Theorem 1.2. Fur- ther, for every r n set ≤

r = A n,d : rk A r and Er = A n,d : rk A = r . E { ∈ M ≤ } { ∈ M }

Given A n,d and i = j, we set ∈ M 6

Fij = Fij(A) := span (Rs)s6=i,j,Ri + Rj . { } Clearly, ker(A) F ⊥. We will be interested in those pairs (i, j) for which this ⊂ ij inclusion turns to equality. Given A n,d, define ∈ M 2 ⊥ KA = (i, j) [n] : i = j and ker A = F (A) .  ∈ 6 ij 2 Lemma 2.4. Let d

T Since y ker A and A , the set I := i : yi = 0 is of cardinality at least ∈ ∈ E1.2 { 6 } n βn. Note that if i I then Ri span Rs, s = i , therefore removing the i-th − ∈ ∈ { 6 } row keeps the rank unchanged, that is, we have rk A = rk Ai, where Ai denotes the n n matrix obtained by substituting the i-th row of A with the zero row. × i i Fix i I. If A n−2 then rk A = rk A n 2. Thus, the non-zero rows of A ∈ ∈E ≤ − are linearly dependent, therefore there exists z Rn 0 such that ∈ \ { } z =0 and z R = 0. i X s s s6=i

Clearly, z ker AT and by the condition A , the set ∈ ∈E1.2

J = J(i) := j n : zj = 0 { ≤ 6 } is of cardinality at least n βn. Note that if j J, then Rj span Rs, s = i, j − ∈ ∈ { 6 } and thus, Ri + Rj span Rs, s = i, j . This means that (i, j) KA. Thus for ∈ { 6 } ∈ A n−2 one has ∈E 2 KA I min J(i) (n βn) . | | ≥ | | i∈I | |≥ −

Now suppose that A En−1 and fix i I. Since Ri span Rs, s = i , there ∈ ∈ ∈ { 6 } exist scalars (xs)s6=i such that

R = x R . i X s s s6=i

T Therefore setting xi = 1, we have x = (xs)s≤n ker A and since A , the − ∈ ∈ E1.2 set L = L(i) := j n : xj = 1 is of cardinality at least n βn. Note that if { ≤ 6 − } − j L, then ∈

Ri + Rj = (xj + 1)Rj + xsRs span Rs, s = i, j X 6∈ { 6 } s6=i,j

(otherwise, we would have Rj span Rs, s = i, j , which is impossible since rk A = ∈ { 6 } n 1). Using again that rk A = n 1, we obtain that dim Fij = n 1, that is, − − − ⊥ dim Fij = 1 = dim ker A.

⊥ Therefore the inclusion ker A F implies that (i, j) KA and the lower bound ⊂ ij ∈ on the cardinality of KA follows.

⊥ Note that for every A n,d the subspace F (A) is invariant under simple ∈ M ij switchings involving the i-th and j-th rows. Moreover, for every pair (i, j) KA ⊥ ∈ one has ker A = Fij (A). Therefore, since our aim is to show that most switchings tend to increase the rank, we need to eliminate those which keep this equality valid, that is, those which keep ker A unchanged. This motivates the following definition.

6 n Definition 2.5. Let d

In the next lemma we estimate the number of x-bad switchings. 2 Lemma 2.6. Let d

Proof. Let λp : p m be the set of disctinct values taken by coordinates of x. { ≤ } For every p m set ≤ Lp = s n : xs = λp . { ≤ } Since A , we have Lp βn for all p m. Since for an x-bad switching in ∈ E1.2 | | ≤ ≤ (i,j,k,ℓ) we have xk = xℓ, k and ℓ should belong to the same Lp. By d-regularity, for every p m the number of switchings in (i,j,k,ℓ) with k,ℓ Lp is at most 2 2 ≤ ∈ d Lp (since we must have aik = ajℓ = 1). Thus, the number of x-bad switchings | | is bounded above by

m m 2 2 2 2 d Lp d max Lp Lp nβnd . X | | ≤ p≤m | | X | |≤ p=1 p=1

3 Proof of Theorem 1.1

We start with the following lemma estimating the number of simple switchings which increase the rank.

Lemma 3.1. Let n 2d be large enough integers and A n−1 1.2. Assume 2 ≥ ∈ E ∩E2 βn := Cn ln d/ ln n n/4. Then there are at least n(n 3βn)d switchings in ≤ − (i,j,k,ℓ) which increase the rank, i.e., for which

rk A¯ = rk A + 1, where A¯ denotes the matrix obtained by the switching.

Proof. Given two rows Ri and Rj, i = j, of A, by d-regularity, there are at most 2 6 d 4-tuples (i,j,k,ℓ) in which a switching can be performed. Thus, the number of c 2 switchings in (i,j,k,ℓ) with (i, j) [n] [n] KA is at most KA d , where the 2 ∈ × \ | | complement is taken in [n] . Therefore, applying Lemmas 2.2 and 2.4 we obtain that the number N of possible switchings in (i,j,k,ℓ) with (i, j) KA is at least ∈ c 2 2 2 2 2 2 2 2 N A K d n(n 2d)d n (n βn) d d n 2dn 2βnn+β . (3) ≥ |F |−| A| ≥ − − − −  ≥ − − n For the rest of the proof, we fix a non-zero vector x ker A. ∈ Fix for a moment (i, j) KA, and note that for any switching on the i-th and ∈ j-th rows, we have ker A¯ F ⊥(A¯)= F ⊥(A) = ker A. ⊂ ij ij 7 Observe that if a switching on i, j-th rows is not x-bad, then Ax¯ = Ax = 0 and 6 therefore ker A¯ = ker A = F ⊥(A¯). 6 ij This means that rk A¯ > rk A and by (1) implies that rk A¯ = rk A + 1. Thus, any possible switching in (i,j,k,ℓ), which is not x-bad and such that (i, j) KA ∈ increases the rank of the matrix by one. Applying Lemma 2.6 and inequality (3), we obtain that the number N0 of switch- ings described above is at least

2 2 2 2 N0 N nβnd d n 2dn 3βnn + β . ≥ − ≥ − − n Since β2 2nd for large enough n, this completes the proof. n ≥ Proof of Theorem 1.1. We may assume that d exp(c√ln n) for a small enough ≤ absolute constant c > 0 (otherwise the probability bound in Theorem 1.1 trivially holds). In this case n 4βn. Fix r 1,...,n 2 and consider the relation ≥ ∈ { − }

Qr (Er ) Er+1, ⊆ ∩E1.2 × defined by (A, A¯) Qr if and only if A Er , A¯ Er+1, and (A, A¯) Q0, ∈ ∈ ∩E1.2 ∈ ∈ where the symmetric relation Q0 is given by (2). Using that any two switchings (i,j,k,ℓ) and (j, i, ℓ, k) produce the same trans- formed matrix, and applying Lemma 3.1 we observe that for every A Er , ∈ ∩E1.2 2 Qr(A) n(n 3βn)d /2. | |≥ −

Now let A¯ Qr(Er ). If A¯ , then by Lemmas 3.1 and 2.2, ∈ ∩E1.2 ∈E1.2 −1 2 2 Q (A¯) ¯ n(n 3βn)d /2 3nβnd /2. | r |≤ |FA|− −  ≤ Otherwise, if A¯ c then ∈E1.2 −1 2 Q (A¯) ¯ /2 n(n d)d /2. | r | ≤ |FA| ≤ − Then Claim 2.1 implies

2 2 2 n(n 3βn)d 3nβnd n(n d)d c − Er Er+1 + − Er+1 . 2 | ∩E1.2|≤ 2 | ∩E1.2| 2 | ∩E1.2| Summing over all r = 1,...,n 2 gives − 3βn n c 3βn c n−2 1.2 n−1 1.2 + n−1 = n−1 + . |E ∩E |≤ n 3βn |E ∩E | n 3βn |E ∩E1.2| n 3βn |E | |E1.2| − − − 2 Using that n 4βn and βn = Cn ln d/ ln n, we obtain ≥ 2 c 12βn c 12C ln d c n−2 n−2 + n−1 + 2 n−1 + 2 . |E |≤|E ∩E1.2| |E1.2|≤ n |E | |E1.2|≤ ln n |E | |E1.2| Theorem 1.2 implies the desired result.

8 Remark 3.2. For A En−1 , Lemma 3.1 guarantees existence of many simple ∈ ∩E1.2 switchings which produce full rank matrices from A. With the above notations, we have 2 Qn−1(A) n(n 3βn)d /2. | |≥ − In order to prove along the same lines that a “typical” matrix in n,d is non- M singular, one needs to consider the reverse operation as well, i.e., to show that for any full rank matrix, there are very few switchings which transform it to a singular one. The argument of this note is based on finding switchings using structural in- formation about vectors in the kernel, specifically, delocalization properties in Theo- rem 1.2. When the matrix is of full rank, we do not have any non-trivial null vectors at hand, which does not allow to revert the above procedure and verify invertibility.

Acknowledgments P.Y. was supported by grant ANR-16-CE40-0024-01. A significant part of this work was completed while the last three named authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, supported by NSF grant DMS-1440140, and the first two named authors visited the institute. The hospitality of MSRI and of the organizers of the program on Geometric Functional Analysis and Applications is gratefully acknowledged.

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Alexander E. Litvak and Nicole Tomczak-Jaegermann, Dept. of Math. and Stat. Sciences, University of Alberta, Edmonton, AB, Canada, T6G 2G1. e-mails: [email protected] and [email protected]

Anna Lytova, Faculty of Math., Physics, and Comp. Science, University of Opole, ul. Oleska 48, 45-052,

10 Opole, Poland. e-mail: [email protected]

Konstantin Tikhomirov, Dept. of Math., Princeton University, Fine Hall, Washington road, Princeton, NJ 08544. e-mail: [email protected]

Pierre Youssef, Universit´eParis Diderot, Laboratoire de Probabilit´es et de mod`eles al´eatoires, 75013 Paris, France. e-mail: [email protected]

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