Department of Mathematics

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This article has two interpenetrating motifs. One is an exposition of some major ideas and techniques behind the use of <a href="https://www.sciencedirect.com/topics/mathematics/block-matrix">block matrices</a>, and especially their positivity <a href="https://www.sciencedirect.com/topics/mathematics/sigma-property">properties</a>. This is done by focussing on one major problem: characterisation of operators whose numerical radius is bounded by one. So, the article could serve as an introduction to that topic as well.

World University root node

Linear Algebra and Its Applications

The numerical radius and positivity of block matrices

Let&nbsp;&lambda;j&nbsp;and&nbsp;ajj,&nbsp;1&le;j&le;n, be the eigenvalues and the diagonal entries of a <a href="https://www.sciencedirect.com/topics/mathematics/hermitian-matrix">Hermitian matrix</a> A, both enumerated in the increasing order. We prove some inequalities that are stronger than the Schur majorisation inequalities&nbsp;&sum;j=1r&lambda;j&le;&sum;j=1rajj,&nbsp;1&le;r&le;n.

Extensions of the Schur majorisation inequalities

Let f be a transformation on the Euclidean space such that f(0) = 0, and f preserves distances. Then f is linear, and this makes it easier to analyze f. The Mazur—Ulam theorem generalizes this to maps between real normed linear spaces. We discuss this theorem and its proofs.

Resonance

Isometries and Linearity

We define the notion of affine Anosov representations of word hyperbolic groups into the affine group&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext mathvariant="sans-serif">SO</mtext><mn>0</mn></msup><mo stretchy="false">(</mo><mi>�</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>�</mi><mo stretchy="false">)</mo><mo>⋉</mo><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">�</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>�</mi><mo>+</mo><mn>1</mn></mrow></msup></math>⁠. We then show that a representation&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>�</mi></math>&nbsp;of a word hyperbolic group is affine Anosov if and only if its linear part&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="monospace">�</mi></mrow><mi>�</mi></msub></math>&nbsp;is Anosov in&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext mathvariant="sans-serif">SO</mtext><mn>0</mn></msup><mo stretchy="false">(</mo><mi>�</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>�</mi><mo stretchy="false">)</mo></math>&nbsp;with respect to the stabilizer of a maximal isotropic plane and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>�</mi><mo stretchy="false">(</mo><mi mathvariant="normal">Γ</mi><mo stretchy="false">)</mo></math>&nbsp;acts properly on&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">�</mi></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>�</mi><mo>+</mo><mn>1</mn></mrow></msup></math>⁠.We define the notion of affine Anosov representations of word hyperbolic groups into the affine group&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext mathvariant="sans-serif">SO</mtext><mn>0</mn></msup><mo stretchy="false">(</mo><mi>�</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>�</mi><mo stretchy="false">)</mo><mo>⋉</mo><msup><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">�</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>�</mi><mo>+</mo><mn>1</mn></mrow></msup></math>⁠. We then show that a representation&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>�</mi></math>&nbsp;of a word hyperbolic group is affine Anosov if and only if its linear part&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="monospace">�</mi></mrow><mi>�</mi></msub></math>&nbsp;is Anosov in&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext mathvariant="sans-serif">SO</mtext><mn>0</mn></msup><mo stretchy="false">(</mo><mi>�</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>�</mi><mo stretchy="false">)</mo></math>&nbsp;with respect to the stabilizer of a maximal isotropic plane and&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>�</mi><mo stretchy="false">(</mo><mi mathvariant="normal">Γ</mi><mo stretchy="false">)</mo></math>&nbsp;acts properly on&nbsp;<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow data-mjx-texclass="ORD"><mi mathvariant="double-struck">�</mi></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>�</mi><mo>+</mo><mn>1</mn></mrow></msup></math>⁠.

International Mathematics Research Notices

Affine Anosov Representations and Proper Actions

In this paper we study the convergence of dynamical discrete web (DyDW) to the dynamical Brownian web (DyBW) in the path space topology. We show that almost surely the DyBW has RCLL paths taking values in an appropriate metric space and as a sequence of RCLL paths, the scaled dynamical discrete web converges to the DyBW. This proves weak convergence of the DyDW process to the DyBW process

Convergence of the dynamical discrete web to the dynamical Brownian web

We establish an analogue of the classical Polya–Vinogradov inequality for 𝐺𝐿(2,𝔽𝑝)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mi>L</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">F</mi></mrow></mrow><mi>p</mi></msub><mo stretchy="false">)</mo></math>, where p is a prime. In the process, we compute the ‘singular’ Gauss sums for 𝐺𝐿(2,𝔽𝑝)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mi>L</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">F</mi></mrow></mrow><mi>p</mi></msub><mo stretchy="false">)</mo></math>. As an application, we show that the collection of elements in 𝐺𝐿(2,ℤ)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mi>L</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">Z</mi></mrow></mrow><mo stretchy="false">)</mo></math>&nbsp;whose reduction modulo p are of maximal order in 𝐺𝐿(2,𝔽𝑝)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mi>L</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">F</mi></mrow></mrow><mi>p</mi></msub><mo stretchy="false">)</mo></math>&nbsp;and whose matrix entries are bounded by x has the expected size as soon as 𝑥≫𝑝1/2+ε<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≫</mo><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mtext>ε</mtext></mrow></mrow></msup></math>&nbsp;for any ε&gt;0<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mtext>ε</mtext></mrow><mo>&gt;</mo><mn>0</mn></math>.

Fulltext

Mathematische Annalen

Singular Gauss sums, Polya–Vinogradov inequality for GL(2) and growth of primitive elements

We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite number-theoretic matrix. We call this matrix the partition-frequency enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-theoretic functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) some well-known recurrence relations for many number-theoretic functions, including the sum of divisors function, Ramanujan’s 𝜏<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi></math>&nbsp;function, sums of squares and triangular numbers, and for 𝜁(2𝑛)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ζ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></math>, where n is a positive integer. These include classical results due to Euler, Ewell, Ramanujan, Lehmer, and others. As one application, we embed Ramanujan’s famous congruences 𝑝(5𝑛+4)≡0(mod5)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo stretchy="false">(</mo><mn>5</mn><mi>n</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo>≡</mo><mn>0</mn><mspace width="thickmathspace">&nbsp;</mspace><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="thinmathspace">&nbsp;</mspace><mn>5</mn><mo stretchy="false">)</mo></math>&nbsp;and 𝜏(5𝑛+5)≡0(mod5)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>τ</mi><mo stretchy="false">(</mo><mn>5</mn><mi>n</mi><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo><mo>≡</mo><mn>0</mn><mspace width="thickmathspace">&nbsp;</mspace><mo stretchy="false">(</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="thinmathspace">&nbsp;</mspace><mn>5</mn><mo stretchy="false">)</mo></math>&nbsp;into an infinite family of such congruences.

The Ramanujan Journal

The partition-frequency enumeration matrix

We study the local eigenvalue statistics ξNω,E<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup></math>&nbsp;associated with the eigenvalues of one-dimensional, (2N+1)×(2N+1)<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></math>&nbsp;random band matrices with independent, identically distributed, real random variables and band width growing as Nα<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>N</mi><mi>α</mi></msup></math>, for 0&lt;α&lt;12<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>. We consider the limit points associated with the random variables ξNω,E[I]<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">]</mo></math>, for I⊂R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>⊂</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math>, and E∈(−2,2)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>∈</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></math>. For random band matrices with Gaussian distributed random variables and for 0≤α&lt;17<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>α</mi><mo>&lt;</mo><mfrac><mn>1</mn><mn>7</mn></mfrac></math>, we prove that this family of random variables has nontrivial limit points for almost every E∈(−2,2)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>∈</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></math>, and that these limit points are Poisson distributed with positive intensities. The proof is based on an analysis of the characteristic functions of the random variables ξNω,E[I]<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">]</mo></math>&nbsp;and associated quantities related to the intensities, as N tends towards infinity, and employs known localization bounds of (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019, Schenker in Commun Math Phys 290:1065–1097, 2009), and the strong Wegner and Minami estimates (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019). Our more general result applies to random band matrices with random variables having absolutely continuous distributions with bounded densities. Under the hypothesis that the localization bounds hold for 0&lt;α&lt;12<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>, we prove that any nontrivial limit points of the random variables ξNω,E[I]<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">]</mo></math>&nbsp;are distributed according to Poisson distributions.We study the local eigenvalue statistics ξNω,E<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup></math>&nbsp;associated with the eigenvalues of one-dimensional, (2N+1)×(2N+1)<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></math>&nbsp;random band matrices with independent, identically distributed, real random variables and band width growing as Nα<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>N</mi><mi>α</mi></msup></math>, for 0&lt;α&lt;12<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>. We consider the limit points associated with the random variables ξNω,E[I]<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">]</mo></math>, for I⊂R<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>⊂</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">R</mi></mrow></math>, and E∈(−2,2)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>∈</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></math>. For random band matrices with Gaussian distributed random variables and for 0≤α&lt;17<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>α</mi><mo>&lt;</mo><mfrac><mn>1</mn><mn>7</mn></mfrac></math>, we prove that this family of random variables has nontrivial limit points for almost every E∈(−2,2)<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>∈</mo><mo stretchy="false">(</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></math>, and that these limit points are Poisson distributed with positive intensities. The proof is based on an analysis of the characteristic functions of the random variables ξNω,E[I]<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">]</mo></math>&nbsp;and associated quantities related to the intensities, as N tends towards infinity, and employs known localization bounds of (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019, Schenker in Commun Math Phys 290:1065–1097, 2009), and the strong Wegner and Minami estimates (Peled et al. in Int. Math. Res. Not. IMRN 4:1030–1058, 2019). Our more general result applies to random band matrices with random variables having absolutely continuous distributions with bounded densities. Under the hypothesis that the localization bounds hold for 0&lt;α&lt;12<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>&lt;</mo><mi>α</mi><mo>&lt;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>, we prove that any nontrivial limit points of the random variables ξNω,E[I]<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mrow class="MJX-TeXAtom-ORD"><mi>ω</mi><mo>,</mo><mi>E</mi></mrow><mi>N</mi></msubsup><mo stretchy="false">[</mo><mi>I</mi><mo stretchy="false">]</mo></math>&nbsp;are distributed according to Poisson distributions.

Journal of Statistical Physics

On the Local Eigenvalue Statistics for Random Band Matrices in the Localization Regime

In this paper, we consider Schrödinger operators on M×Zd2<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>×</mo><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">Z</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub></mrow></msup></math>, with M={M1,…,M2}d1<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mo>=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>M</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>M</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><msup><mo fence="false" stretchy="false">}</mo><mrow class="MJX-TeXAtom-ORD"><msub><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub></mrow></msup></math>&nbsp;(‘quantum wave guides’) with a ‘Γ<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Γ</mi></math>-trimmed’ random potential, namely a potential which vanishes outside a subset Γ<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Γ</mi></math>&nbsp;which is periodic with respect to a sub-lattice. We prove that (under appropriate assumptions) for strong disorder these operators have pure point spectrum outside the set Σ0=σ(H0,Γc)<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mtext>Σ</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mtext>σ</mtext></mrow><mo stretchy="false">(</mo><msub><mi>H</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>,</mo><msup><mi mathvariant="normal">Γ</mi><mrow class="MJX-TeXAtom-ORD"><mi>c</mi></mrow></msup></mrow></msub><mo stretchy="false">)</mo></math>&nbsp;where H0,Γc<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>H</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn><mo>,</mo><msup><mi mathvariant="normal">Γ</mi><mrow class="MJX-TeXAtom-ORD"><mi>c</mi></mrow></msup></mrow></msub></math>&nbsp;is the free (discrete) Laplacian on the complement Γc<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">Γ</mi><mrow class="MJX-TeXAtom-ORD"><mi>c</mi></mrow></msup></math>&nbsp;of Γ<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Γ</mi></math>. We also prove that the operators have some absolutely continuous spectrum in an energy region E⊂Σ0<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">E</mi></mrow></mrow><mo>⊂</mo><msub><mrow class="MJX-TeXAtom-ORD"><mtext>Σ</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>. Consequently, there is a mobility edge for such models. We also consider the case −M1=M2=∞<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><msub><mi>M</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>M</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">∞</mi></math>, i.e.&nbsp; Γ<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Γ</mi></math>-trimmed operators on Zd=Zd1×Zd2<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">Z</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow></msup><mo>=</mo><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">Z</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub></mrow></msup><mo>×</mo><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">Z</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><msub><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub></mrow></msup></math>. Again, we prove localisation outside Σ0<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow class="MJX-TeXAtom-ORD"><mtext>Σ</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>&nbsp;by showing exponential decay of the Green function GE+iη(x,y)<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mi>E</mi><mo>+</mo><mi>i</mi><mi>η</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math>&nbsp;uniformly in η&gt;0<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi><mo>&gt;</mo><mn>0</mn></math>. For all energies E∈E<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>∈</mo><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">E</mi></mrow></mrow></math>&nbsp;we prove that the Green’s function GE+iη<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mrow class="MJX-TeXAtom-ORD"><mi>E</mi><mo>+</mo><mi>i</mi><mi>η</mi></mrow></msub></math>&nbsp;is not (uniformly) in ℓ1<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>ℓ</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msup></math>&nbsp;as η<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi></math>&nbsp;approaches 0. This implies that neither the fractional moment method nor multi-scale analysis can be applied here.