the empirical eigenvalue measure of a large random matrix is typically well approximated by a deterministic asymptotic measure .for instance , for wigner matrices this measure is the celebrated wigner semicircle law .this approximation is best formulated using the green function and stieltjes transforms .let be an hermitian random matrix normalized so that its typical eigenvalue spacing is of order , and denote by the associated green function . here is a spectral parameter with positive imaginary part .then the stieltjes transform of the empirical eigenvalue measure is equal to , and the approximation mentioned above may be written informally as for large and with high probability . here is the stieltjes transform of the asymptotic eigenvalue measure , which we denote by .we call an estimate of the form an _ averaged law_.  as may be easily seen by taking the imaginary part of , control of the convergence of yields control of an order eigenvalues around the point . a_ local law _ is an estimate of the form for all .note that the approximation can not be correct at or below the scale , at which the behaviour of the left - hand side of is governed by the fluctuations of individual eigenvalues .local laws have become a cornerstone of random matrix theory , starting from the work where a local law was first established for wigner matrices .local laws constitute the main tool needed to analyse the distribution of the eigenvalues of random matrices , including the universality of the local spectral statistics and the proof of the celebrated wigner - dyson - mehta conjecture , and the eigenvectors of random matrices , including eigenvector delocalization and the distribution of the eigenvector components .in fact , for the aforementioned applications to the distribution of eigenvalues and eigenvectors , the averaged local law from is not sufficient .one has to control not only the normalized trace of but the matrix itself , by showing that is close to some deterministic matrix , , provided that .as the dimension of these matrices is very large , which notion of closeness to use is a nontrivial question .the control of as a matrix was initiated in in the context of wigner matrices , where the individual matrix entries are controlled .we call such a result an _ entrywise local law_. subsequently , the entrywise local law of has been refined and significantly generalized .  more generally , canonical notions are closeness in the _ weak operator sense _ , _ strong operator sense _ , and _norm sense _ , which entail control of respectively , for deterministic .it is easy to see that already convergence in the strong operator sense must fail , as one typically has ; we refer to ( * ? ? ?* section 12.1 ) for more details .we conclude that control in the weak operator sense , i.e.  of , is the strongest form of control that one may expect to hold for the error .following we call such a result an _ ( an)isotropic local law_. ( an)isotropic local laws have played a central role in the study of deformed matrix ensembles and the distribution of eigenvector components of random matrices .all proofs of local laws consist of at least the two following major steps , which are completely decoupled .  1 .  _ a stochastic step _ establishing a self - consistent equation for with a random error term .2 .  _ a deterministic step _ analysing the stability of the self - consistent equation .  step ( a ) may be formulated in general as follows .there is a map such that is uniquely charaterized as the solution of the equation with positive imaginary part . for a large class of random matrices , including those considered in this paper, is a nonlinear quadratic function .step ( a ) provides high - probability bounds on the matrix ( in some suitable sense ) .  in step ( b )one shows that if is small then so is .  in general ,both steps are nontrivial .in addition , steps ( a ) and ( b ) have up to now only been used to derive an entrywise local law ; to obtain a full ( an)isotropic local law , a further nontrivial argument was necessary ; three different techniques for deriving ( an)isotropic laws from entrywise laws were developed in .  in this paperwe develop a new method for dealing with step ( a ) , which has the following advantages as compared to previous approaches .  1.  it can handle very general mean - field random matrices , and is in particular completely insensitive to the expectation .( previous methods heavily rely on the assumption that is diagonal or close to diagonal . ) 2 .it automatically yields the ( an)isotropic law in one step , without requiring a further intermediate step via an entrywise law , as in .3 .  it is simple and very versatile . it is applicable to many matrix models where previous methods either fail or are very cumbersome .  for conciseness , in this paperwe focus on mean - field hermitian random matrices whose upper - triangular entries are independent . by _mean - field _ we mean that for all .we make no assumption on .we remark that our method extends to very general matrices built from independent random variables , such as sample covariance matrices built from populations with arbitrary expectations and covariances .we now outline our main results . throughout the paper we split define the map \,.\ ] ] then it is not hard to show that for , the equation has a unique solution with positive imaginary part ( see lemma [ lem : sol_pi ] below ) .our main result ( theorem [ thm : main_general ] below ) is the derivation of the self - consistent equation : we establish high - probability bounds on the quantities \,,\ ] ] where is a bounded deterministic matrix and are deterministic .these bounds are sharp ( up to some technicalities in the definition of a high probability bound ) throughout the spectrum , in both the bulk and in the vicinity of regular edges where the density of the limiting measure exhibits the characteristic square - root decay of random matrix theory .we emphasize that we make no assumption on the expectation .our main result performs step ( a ) for the general class of mean - field random matrices introduced above . to illustrate how this resultmay be combined with step ( b ) to complete the proof of the ( an)isotropic local law , we use our main result to settle in complete generality the local law for wigner matrices with arbitrary expectation ( theorem [ thm : wigner ] below ) , i.e.  with arbitrary expectation .previously , the local law for wigner matrices with nonzero expectation was derived under the assumption that is diagonal or that all third moments of vanish .even the averaged local law for general was not accessible using previous methods .  in this paper we do not perform step ( b ) ( the stability analysis of the self - consistent equation ) for the aforementioned general class of mean - field matrices . for general is a nontrivial task , and we refer to the works , where a related analysis is performed to obtain an entrywise law for the case of almost diagonal .we remark that the map defined in has been extensively studied in the random matrix literature , starting with , where its stability was analysed on the global spectral scale .recently , it has been the focus of the seminal series of works mentioned above , where the stability of the matrix equation was analysed in great detail in order to establish local laws for a general class of random matrix ensembles .we conclude this introductory section with a few words about the proof .traditionally , in all of the works , the proof of step ( a ) for matrices with independent entries relied on schur s complement formula .when applicable , schur s complement formula provides a direct approach to proving step ( a ) .however , its applicability to random matrix models is heavily dependent on the independence of the entries of and on the fact that is diagonal .if the mapping between the independent random variables and the entries of is nontrivial or if is not diagonal , the use of schur s complement formula becomes more cumbersome and possibly even fruitless .moreover , schur s complement formula is only effective when deriving entrywise estimates , and it is ill - suited for deriving isotropic estimates .the first proof of a local law without using schur s complement formula was , where a local law was established for random regular graphs ( which are not built from independent random variables ) . in ,schur s complement formula was replaced by a much more robust expansion using the resolvent identity .our current approach is in part motivated by this philosophy : instead of working with entire rows and columns , as dictated by schur s complement formula , we work with individual matrix entries . a natural way of achieving this is to perform a series of expansions using the resolvent identity , where is expanded in a single entry of .we remark that such ideas have previously been used in the somewhat different context of proving the so - called fluctuation averaging result in random matrix theory ; see .  in practice, we choose to replace the resolvent expansion with a cumulant expansion inspired by ( see lemma [ lem : cumulant_expansion ] below ) , which effectively performs the same steps but in a more streamlined fashion .a drawback of the cumulant expansion is that it requires working with expectations of random variables instead of just random variables .this leads us to consider high moments of error terms , which are estimated self - consistently .an analogous strategy was first used in to prove the isotropic local law for wigner matrices , although there the moments were estimated by the green function comparison strategy instead of the cumulant expansion used in this paper .the first use of recursive self - consistent estimates for high moments using the cumulant expansion , including exploiting a cancellation among two leading terms , was ; this strategy constitutes the core of our method .the cumulant expansion was first used to derive a local law in the recent work , where a precise averaged law was derived for sparse random matrices with .analogous ideas for the haar measure on the unitary group were recently exploited in to obtain optimal bounds on the convergence rate for the spectral distribution of the sum of random matrices . historically , the first use of the cumulant expansion for local spectral statistics is , where edge universality and the tracy - widom limit for deformed wigner matrices was proved , along with a streamlined proof of the edge universality for wigner matrices .  aside from the novel idea of using the resolvent / cumulant expansion to prove ( an)isotropic local laws for random matrices with independent entries ,the key new ingredients of our method may be summarized as follows .  1 .  we bootstrap isotropic bounds on the green function , i.e. our proof makes use of a priori bounds of the form with high probability . when using the cumulant expansion ( or , alternatively , the resolvent expansion ) , we are naturally led to sums of the form .it is crucial to first perform the sum over to exploit the isotropic a priori bounds on ; simply estimating the sum term by term leads to errors which are not affordable .reducing such sums to isotropic estimates on the green function requires a certain amount of care .2 .  a direct application of the cumulant expansion to estimating high moments of the error results in terms that can not be controlled with a high enough precision . to circumvent this issue , we use multiple self - improving matrix bounds at several stages of the proof .we assume a rough a priori bound on an error matrix and derive a self - improving bound on it , which may be iterated to obtain the optimal bound .see e.g.  or below for an example . for the estimate of ] for all and .let be a spectral parameter , and define the green function we use the following notion of high - probability bounds , the first version of which appeared in .[ def : stocdom]  1 .let be two families of nonnegative random variables , where is a possibly -dependent parameter set .+ we say that is _ stochastically dominated by , uniformly in , _ if for all ( small ) and ( large ) we have for large enough . throughout this paperthe stochastic domination will always be uniform in all parameters ( such as matrix indices , deterministic vectors , and spectral parameters ) that are not explicitly fixed .note that may depend on quantities that are explicitly constant .if is stochastically dominated by , uniformly in , we use the notation . moreover ,if for some complex family we have we also write .( note that for deterministic and , means for any . )we extend the definition of to matrices in the weak operator sense as follows .let be a family of complex random matrices and a family of nonnegative random variables .then we write to mean uniformly for all deterministic vectors .[ def_s_f ] define and .let be a linear map that maps to itself . for define the function through   the following deterministic result was proved in .[ lem : sol_pi ] let and be as in definition [ def_s_f ] .then for any the equation has a unique solution .we denote this solution by .we now choose to be \,.\ ] ] fix a constant and define the fundamental domain our first main result gives optimal high - probability bounds on .[ thm : main_general ] suppose that assumption [ ass : h ] holds .denote by the solution of with respect to the linear map .let and suppose that and for some deterministic  ] and denote by the unit sphere of . for an matrix , vectors , and \!]} ] .if the above random variables depend on some parameter and the hypotheses are uniform in then so are the conclusions .the proof follows from the definition of stochastic domination together with a union bound argument .[ lem : h_bound ] under assumption [ ass : h ] we have .this is a standard application of the fredi - komls argument ( see e.g. ( * ? ? ?* section 2.1.6 ) ) .[ [ ward - identity ] ] ward identity + + + + + + + + + + + + +  an important ingredient of the proofs is the following well - known identity .[ lem : ward_identity ] for every vector   this follows immediately from the resolvent identity .  as a consequence , for any deterministic we havethe estimate   [ [ cumulant - expansion ] ] cumulant expansion + + + + + + + + + + + + + + + + + +  recall that for a real random variable , all whose moments are finite , the -cumulant of is \bigg)\bigg|_{t=0}.\ ] ] we shall use a standard cumulant expansion from , which we record in lemma [ lem : cumulant_expansion ] below .we require slightly stronger bounds on the remainder term than in previous applications , and for completeness we give a proof of lemma [ lem : cumulant_expansion ] in appendix [ a ] .[ lem : cumulant_expansion ] let be a smooth function , and denote by its -th derivative . then , for every fixed , we have =\sum_{k=0}^{\ell}\frac{1}{k!}\mathcal{c}_{k+1}(h)\mathbb{e}[f^{(k)}(h)]+{\mathcal}r_{\ell+1},\ ] ] assuming that all expectations in exists , where is a remainder term ( depending on and ) , such that for any ,   the following result gives bounds on the cumulants of the entries of .[ lem : cumulant_factos_estimate ] if satisfies assumption [ ass : h ] then for every \!]} ] .suppose there exists a constant such that for any  ] denote \ ! ] } } \ , , \qquad \text { where } \quad x^{j}_i { \mathrel{\vcenter{\baselineskip0.65ex \lineskiplimit0pt \hbox{.}\hbox{.}}}=}x_i { s}_{ij}\,.\ ] ] we define the set of vectors note that , due to assumption [ ass : h ] , we have , so that for all and \!]} ] .2 .  for , .3 .  for any there exists such that uniformly for all \!]} ] since .hence , \\   & = -\sum_{i , j}{v}_i{\mathbb{e}}[h_{ij}^2]\cdot{\mathbb{e}}\big[(g_{ji}g_{j{\boldsymbol{\mathrm{w}}}}+g_{jj}g_{i{\boldsymbol{\mathrm{w}}}})(1+\delta_{ij})^{-1}{\mathcal}d_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}^{p-1}\overline{{\mathcal}d}_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}^p\big ]   + \sum_{i , j}v_i{\mathbb{e}}\big[h_{ij}^2\big]{\mathbb{e}}\big[g_{j{\boldsymbol{\mathrm{w}}}}\partial_{ij}\big({\mathcal}d_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}^{p-1}\overline{{\mathcal}d}_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}^p\big)\big]\\   & = -\mathbb{e}\big[(\mathcal{s}(g)g))_{\mathbf{v}\mathbf{w}}\mathcal{d}_{\mathbf{v}\mathbf{w}}^{p-1}\overline{\mathcal{d}}_{\mathbf{v}\mathbf{w}}^{p}\big]+ \sum_{i , j}{v}_i{\mathbb{e}}\big[h_{ij}^2\big]{\mathbb{e}}\big[g_{j{\boldsymbol{\mathrm{w}}}}\partial_{ij}\big({\mathcal}d_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}^{p-1}\overline{{\mathcal}d}_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}^p\big)\big]\ , ,   \end{aligned}\ ] ] and consequently , recalling , we have +x_1   = -{\mathbb{e}}[{\mathcal}j_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}{\mathcal}d_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}^{p-1}\overline{{\mathcal}d}_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}^p]+\sum_{i , j}{v}_i{\mathbb{e}}[h_{ij}^2]{\mathbb{e}}[g_{j{\boldsymbol{\mathrm{w}}}}\partial_{ij}\big({\mathcal}d_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}^{p-1}\overline{{\mathcal}d}_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}^p\big)]\,.\ ] ] since implies = o_\prec ( \hat \zeta ) \cdot{\mathbb{e}}|{\mathcal}d_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}|^{2p-1} ] , and hence the desired estimate follows from the assumption .  what remains , therefore , is to estimate for , which we assume from now on . because by assumption , we find that . moreover , since , we find that .thus , it remains to consider the three cases , and .we deal with them separately .[ [ the - case - rq0k ] ] the case + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  in this case , so that reads where we used lemma [ 3.1 ] and the assumption . by , ,the ward identity , the bound , and the assumption we have thus , we expand the square on the right - hand side of and estimate the result term by term , using , the ward identity , and .for example we have and where in the last step we used the estimates and . by estimating other terms on the right - hand side of in a similar fashion , we get the bound together with we therefore conclude that desired estimate then follows from the assumptions and .[ [ the - case - rq1k-1 ] ] the case + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  we apply a similar argument to the one from the previous case . in this case , and reads where in the second step we used lemma [ 3.1 ] and the assumption . using and to rewrite and one factor of respectively , we have we expand the right - hand side of the above and estimate the result term by term , using , the ward identity , and .for example we have and by estimating the other terms on the right - hand side of in a similar fashion , we get the bound together with we conclude that the desired estimate then follows from the assumptions and .  [ [ the - case - rq0k-1 ] ] the case + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +  since and for every , there exists exactly one such that and the remaining s are .hence , reads where we used lemma [ 3.1 ] . bywe have where is defined as in .now we apply the differential on the right - hand side , and estimate the result term by term . by using the bounds and , the ward identity , and in a similar fashion as in the previous two cases , we get together withwe conclude that desired estimate then follows from the assumptions and .this concludes the proof .the remainder term of the cumulation expansion was analysed in a slightly different context in , and we shall follow the same method .let be given .fix , and choose in .define , where we recall the notation .then we have .let .we have the resolvent expansions and note that only two entries of are nonzero , and they are stochastically dominated by .then the bound together with , , and the assumption , show that combining with , the trivial bound , and the fact is independent of , we have then a simple induction using implies for every fixed . also , by assumption [ ass : h ] we have a trivial bound \!]}}\sup_{|h_{ij}|{\leqslant}t}|h_{ab}| \prec 1\,.\ ] ]  now let us estimate the second term on the right - hand side of . by our definition of in , together with , , and , one easily shows that for any fixed .note that , and by the bound we can find such that } = o(n^{-d})\,.\ ] ]  finally , we estimate the first term on the right - hand side of . for \!]} ] . together with assumption [ ass : h ] , , , and the trivial bound , we have from assumption [ ass : h ] we find , thus by cauchy - schwarz inequality we have a combination of , and shows that the first term on the right - hand side of is bounded by .this completes the proof of , and our steps also show that the bound is uniform for all \!]} ] .2 .  for , .3 .  for any , there exists such that uniformly for all \!]} ] , the formula for the derivative of ( see ) , and the one for the derivative of ( see ) , one can verify that +y_1 ) \\   &\qquad = -(1+\delta_{ij}){\mathbb{e}}[\underline{{\mathcal}j b}\cdot  \underline{{\mathcal}d b}^{p-1 } \overline{\underline{{\mathcal}d b}}^p ] \\   &\qquad + \frac{(p-1)}{n^3}\sum_{i , j}s_{ij}{\mathbb{e}}\big[((gb)_{ji}(gb)_{ij}+(gb)_{ji}(gb)_{ji})\underline{{\mathcal}d b}^{p-2}\overline{\underline{{\mathcal}d b}}^p\big]\\   & \qquad + \frac{p}{n^3}\sum_{i , j}s_{ij}{\mathbb{e}}\big[((gb)_{ji}(\overline{gb})_{ij}+(gb)_{ji}(\overline{gb})_{ji})\underline{{\mathcal}d b}^{p-1}\overline{\underline{{\mathcal}d b}}^{p-1}\big]\\   & \qquad - \frac{p-1}{n^2 } \sum_{i , j } s_{ij } { \mathbb{e}}\big[(gb)_{ji}(q_{ij}+q_{ji})\underline{{\mathcal}d b}^{p-2}\overline{\underline{{\mathcal}d b}}^p\big]\\   & \qquad - \frac{p}{n^2 } \sum_{i , j } s_{ij } { \mathbb{e}}\big[(gb)_{ji}(\overline{q}_{ij }   + \overline{q}_{ji})\underline{{\mathcal}d b}^{p-1}\overline{\underline{{\mathcal}d b}}^{p-1}\big]\ , .   \end{aligned}\ ] ] note that on the right - hand side of this identity a crucial cancellation took place : the first line comes from the sum of the term and the term in , which almost cancel , yielding the small term .  taking an absolute value , using for the first term ,the ward identity and the fact that for the second and third terms and lemma [ lem : estimation_of_q ] ( ii ) together with the ward identity for the last two terms , the claim follows .   from lemma [ lem : cumulant_factos_estimate ]we get , and therefore as before , we ignore the complex conjugates to simplify notation , and estimate instead of .the proof is performed using an explicit computation of the second derivative in self - explanatory notation .starting with , we have using the bound , the ward identity , and on each of the four terms , we obtain as desired .the first term of yields the contribution where we used . using , , the bound , and the ward identity, one can estimate the contribution of the first two terms by .using , lemma [ lem : estimation_of_q ] ( ii ) , and the ward identity , one can bound the contribution of the last two terms by , as desired .in the same spirit , using lemma [ lem : estimation_of_q ] ( i ) , the second term of is of the form   finally , the contribution of is of the form using lemma [ lem : estimation_of_q ] ( i ) to bound and using the ward identity for , we conclude that the last term equals , as desired .this concludes the proof of lemma [ prop : main_estimations_averaging ] ( i ) and ( ii ) .the proof is similar to that of lemma [ prop : main_estimations ] ( iii ) , and we omit the details .we start with several observations regarding the matrix .define the random matrix thus we have and therefore by we have for all  where .we prove the claim by induction on .we start with the case .fix .using , , and the ward identity , we obtain using the bound the ward identity , and lemma [ lem : h_bound ] , we therefore get similarly , using the assumption we find .we conclude that , which completes the proof of the case .we now perform the induction step .suppose that and for .by we have we deal with each of the terms on the right - hand side separately .the term is estimated using as where in the third step we used the induction assumption .the three subsequent terms of are estimated analogously .  in order to estimate the last term of, we write using , the assumption and the bound it follows that what remains , therefore , is to estimate applied to the four last terms of .they are all similar , and we consider for definiteness . by computing the derivatives and using , we obtain a sum of terms , each of which is a product of factors of the form for some and one factor of the form for some .furthermore , exactly two of those factors are of the form or with . applying the ward identity with the sum over to the two factors containing an index , and estimating the remaining factors using, we conclude that as desired .this concludes the proof .[ proof of lemma [ lem : estimation_of_q ] ( ii ) ] we have to improve the naive bound from part ( i ) by an order .we do this by deriving a stochastic self - improving estimate on , which makes use of a crucial cancellation analogous to , but more subtle than , the one in .we derive this self - improving bound using another high - moment estimate for the entries of , which is derived using the cumulant expansion from lemma [ lem : cumulant_expansion ] .  to thisend , fix and . fromwe immediately get thus , + \frac{1}{n}\sum_{i , j}{\mathbb{e}}\big[(gb)_{{\boldsymbol{\mathrm{v } } } i}h_{ij}g_{j{\boldsymbol{\mathrm{w}}}}q_{{\boldsymbol{\mathrm{v } } } { \boldsymbol{\mathrm{w}}}}^{p-1}\overline{q}_{{\boldsymbol{\mathrm{v } } } { \boldsymbol{\mathrm{w}}}}^{p}\big]\,,\ ] ] where we defined using the cumulant expansion ( lemma [ lem : cumulant_expansion ] ) for the last term on the right - hand side with and , we obtain + \sum_{k=1}^{\ell } z_k + \sum_{i , j } \hat{{\mathcal}r}_{\ell+1}^{ij}\ , , \ ] ] where we used the notation \ ] ] here is a fixed positive integer to be chosen later , and is a remainder term defined analogously to in ( [ tau1 ] ) .  the proof of lemma [ lem : estimation_of_q ] can be broken into the following estimates .[ lem : cumulant_expansion_for_q_estimates ] suppose that for some .  1 .we have +z_1 = o_\prec(\zeta^3 ) { \mathbb{e}}|q_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}|^{2p-1}+o_\prec(\zeta^3 \lambda){\mathbb{e}}|q_{{\boldsymbol{\mathrm{v}}}{\boldsymbol{\mathrm{w}}}}|^{2p-2}\ ] ] 2 .for we have 3 .  for any , there exists such that uniformly for all \!]} ] such that is decreasing for each .  for a control parameter define the following result is a simple consequence of theorem [ thm : main_general ] applied to the special case of wigner matrices .[ lem : main_est_wig ] suppose that the assumptions of theorem [ thm : wigner ] hold .let be a control parameter .let and suppose that at have at  2 .  for any deterministic  we have at   to prove ( i ) , by lemmas [ lem : m_eta ] and [ lem : imm ] , it suffices to prove where denotes the map from . bywe have , and shows . also , assumption [ ass : wigner ] shows , and we obtain from that for any fixed , where in the last step we used , , and the ward identity .this proves the first estimate of .the second estimate of is proved similarly .the proof of ( ii ) is analogous , using the estimates and where in the last step we used lemma [ lem : ward_identity ] and the estimate , and lemmas [ lem : m_eta ] and [ lem : imm ] .this concludes the proof .for define with this notation , we have taking the difference yields moreover , taking the trace of the second identity of yields where we defined note that , by .the formulas and are the main identities behind the following analysis .  in order to estimate the error terms on the right - hand side of and using lemma [ lem : main_est_wig ] , we need to deal with the fact that the matrix is random .an event _ holds with high probability _ if , i.e. if for all and large enough depending on .[ lem : rg_est ] suppose that the assumptions of lemma [ lem : main_est_wig ] hold .suppose that with high probability .then the conclusions ( i ) and ( ii ) of lemma [ lem : main_est_wig ] hold with replaced by .let an event of high probability such that , and define the set . since , we find that is contained in the ball of radius around the origin .let be an net of , i.e. a set such that and for each there exists a such that .  by a union bound , from lemma [ lem : main_est_wig ]( i ) it follows that for any  writing and estimating the first term by using and the second term by ( since trivially ) , we conclude that .here we used that by lemma [ lem : m_eta ] .the quantity is estimated analogously .this concludes the proof .we begin by estimating for large .[ lem : large_eta ] under the assumptions of theorem [ thm : wigner ] , we have for .  for have and therefore .moreover , we have and . from , , and lemma [ lem : rg_est ]we therefore find for we have , and the claim follows since is a probability measure .[ lem : g - m_iter ] suppose that at we have , , and for some control parameter .then we have at   from the resolvent identity and the assumptions and , we conclude that with high probability .now suppose that for some control parameter .then from and lemma [ lem : rg_est ] we find using that with high probability , we deduce the implication the implication is a self - improving bound , which we iterate to obtain successively better bounds on .the iteration gives rise to a sequence , where and . by the definition of , we easily conclude the claim after a bounded number of iterations .note that here we used the fact that in order to make sure that , and for theorem [ thm : main_general ] , and thus lemma [ lem : main_est_wig ] , to hold .see also lemma [ lem : self - improv ] and its proof .lemma [ lem : g - m_iter ] provides a bound on starting from a bound on and a rough bound on .the needed bound on is obtained from a stochastic continuity argument , which propagates the bound on from large values of to small values of .it makes use of the following notion of stability of the equation .[ def : stability ] let be a spectral domain .we say that is _stable on _ if the following holds for some constant .suppose that the line  ] be continuous such that is decreasing on  ] for .here .the stochastic continuity argument is an induction on .we start with . by lemmas [ lem : large_eta ] and [lem : g - m_iter ] with we have and for , since trivially .the induction step follows from the following result .fix and suppose that and hold for all . then and hold for all .denote by the upper edge of the line .first we note that by a simple monotonicity of the resolvent ( see e.g. ( * ? ? ?* lemma 10.2 ) ) , the estimate for implies for . setting in lemma [ lem : rg_est ], we find from that for all .using that is -lipschitz in , we find using a simple -net argument that with high probability we have for all ; here we also used that by definition of . moreover ,from , we find that with high probability we have for . from definition [ def : stability ]we therefore deduce that   next , suppose that is a control parameter and for .in particular , from we deduce that with high probability for all . from lemma [ lem : g - m_iter ]we deduce that . by lemma [ lem : rg_est ] with and we therefore have it is easy to verify that is a control parameter .moreover , by the induction assumption we have from an -net argument on analogous to the one given in the previous paragraph , we conclude using definition [ def : stability ] that for all we have in conclusion , for every we have the implication starting from and iterating this implication a bounded number of times and using the definition of , we obtain .see lemma [ lem : self - improv ] and its proof. now follows from lemma [ lem : g - m_iter ] .this concludes the proof .we have established the claim of theorem [ thm : wigner ] on the whole line . since was arbitrary , this concludes the proof of theorem [ thm : wigner ] .in this section we prove the cumulant expansion formula with the remainder term . we start with an elementary inequality .it suffice to assume .let us abbreviate . for , we have {\leqslant}{\mathbb{e}}\big [ t^ax^b \mathbf{1}_{x > t}\big]{\leqslant}{\mathbb{e}}\big[x^{a+b}\mathbf{1}_{x > t}\big]\,.\ ] ] for , we have >{\mathbb{e}}\big[t^ax^b \mathbf{1}_{x{\leqslant}t}\big ] { \geqslant}{\mathbb{e}}\big[x^{a+b}\mathbf{1}_{x{\leqslant}t}\big]\,,\ ] ] and the claim follow from .  fix . by taylor expansion we can find a polynomial of degree at most , such that for any , where is a random variable taking value between 0 and . also , it is easy to verify that =\sum_{k=0}^{\ell } \frac{1}{k!}{\mathcal}c_{k+1}(h){\mathbb{e}}\big [ p^{(k)}(h)\big]\,.\ ] ] by , , homogeneity of the cumulants , and jensen s inequality we have -\sum_{k=0}^{\ell}\frac{1}{k!}\mathcal{c}_{k+1}(h)\mathbb{e}\big[f^{(k)}(h)\big]\\ & = \mathbb{e}\big[h\cdot ( f(h)-p(h))\big]-\sum_{k=0}^{\ell}\frac{1}{k!}\mathcal{c}_{k+1}(h)\mathbb{e}\big[f^{(k)}(h)-p^{(k)}(h)\big]\\ & = \frac{1}{(\ell+1)!}{\mathbb{e}}\big[f^{(\ell+1)}(\xi_0)\cdot h^{\ell+2}\big]-\sum_{k=0}^{\ell}\frac{1}{k!(\ell+1-k)!}\mathcal{c}_{k+1}(h)\mathbb{e}\big[f^{(\ell+1)}{(\xi_k)}\cdot h^{\ell+1-k}\big]\\ & { \leqslant}o(1)\cdot \sum_{k=0}^{l+1 } { \mathbb{e}}|h|^k \cdot { \mathbb{e}}\big[\sup_{|x|{\leqslant}|h|}\big|f^{(\ell+1)}(x)\big|\cdot h^{\ell+2-k } \big]\\ & { \leqslant}o(1)\cdot { \mathbb{e}}|h|^{\ell+2}\cdot \sup_{|x|{\leqslant}t}\big|f^{(l+1)}(x)\big| + o(1)\cdot \sum_{k=0}^{l+1 } { \mathbb{e}}|h|^k \cdot { \mathbb{e}}\big[\sup_{|x|{\leqslant}|h|}\big|f^{(\ell+1)}(x)\big|\cdot h^{\ell+2-k } \cdot \mathbf{1}_{|h|>t}\big]\ , .\end{aligned}\ ] ] the desired result then follows from estimating the last term of by cauchy - schwarz inequality and lemma [ jeson ] .

Abstract: we present a simple and versatile method for deriving ( an)isotropic local laws for general random matrices constructed from independent random variables . our method is applicable to mean - field random matrices , where all independent variables have comparable variances . it is entirely insensitive to the expectation of the matrix . in this paper we focus on the probabilistic part of the proof the derivation of the self - consistent equations . as a concrete application , we settle in complete generality the local law for wigner matrices with arbitrary expectation .