Wigner random matrices with non-symmetrically distributed entries

We show that the spectral radius of an $N\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from above by 2 \*\sigma + o(N^{-6/11+\epsilon}), where $\sigma^2$ is the variance of the matrix entries and $\epsilon$ is an arbitrary small positive number. Our bound improves the earlier results by Z.F\"{u}redi and J.Koml\'{o}s (1981), and the recent bound obtained by Van Vu (2005).Comment: to appear in the Special Issue on Random Matrices of the Journal of Statistical Physic

On the lower bound of the spectral norm of symmetric random matrices with independent entries

We show that the spectral radius of an $N\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from below by 2 \*\sigma - o(N^{-6/11+\epsilon}), where $\sigma^2$ is the variance of the matrix entries and $\epsilon$ is an arbitrary small positive number. Combining with our previous result from [7], this proves that for any $\epsilon >0,$ one has \|A_N\| =2 \*\sigma + o(N^{-6/11+\epsilon}) with probability going to 1 as $N \to \infty.

Universality of slow decorrelation in KPZ growth

There has been much success in describing the limiting spatial fluctuations of growth models in the Kardar-Parisi-Zhang (KPZ) universality class. A proper rescaling of time should introduce a non-trivial temporal dimension to these limiting fluctuations. In one-dimension, the KPZ class has the dynamical scaling exponent $z=3/2$, that means one should find a universal space-time limiting process under the scaling of time as $t\,T$, space like $t^{2/3} X$ and fluctuations like $t^{1/3}$ as $t\to\infty$. In this paper we provide evidence for this belief. We prove that under certain hypotheses, growth models display temporal slow decorrelation. That is to say that in the scalings above, the limiting spatial process for times $t\, T$ and $t\, T+t^{\nu}$ are identical, for any $\nu<1$. The hypotheses are known to be satisfied for certain last passage percolation models, the polynuclear growth model, and the totally / partially asymmetric simple exclusion process. Using slow decorrelation we may extend known fluctuation limit results to space-time regions where correlation functions are unknown. The approach we develop requires the minimal expected hypotheses for slow decorrelation to hold and provides a simple and intuitive proof which applied to a wide variety of models.Comment: Exposition improved, typos correcte

Poisson convergence for the largest eigenvalues of Heavy Tailed Random Matrices

We study the statistics of the largest eigenvalues of real symmetric and sample covariance matrices when the entries are heavy tailed. Extending the result obtained by Soshnikov in \cite{Sos1}, we prove that, in the absence of the fourth moment, the top eigenvalues behave, in the limit, as the largest entries of the matrix.Comment: 22 pages, to appear in Annales de l'Institut Henri Poincar

Universality results for largest eigenvalues of some sample covariance matrix ensembles

For sample covariance matrices with iid entries with sub-Gaussian tails, when both the number of samples and the number of variables become large and the ratio approaches to one, it is a well-known result of A. Soshnikov that the limiting distribution of the largest eigenvalue is same as the of Gaussian samples. In this paper, we extend this result to two cases. The first case is when the ratio approaches to an arbitrary finite value. The second case is when the ratio becomes infinity or arbitrarily small.Comment: 3 figures 47 pages Simulations have been included, a mistake in the computation of the variance has been corrected (Section 2.5

Evidence of functional specificity within the MAGE-I family of tumor expressed proteins

The Melanoma Antigen Genes (MAGE) belong to a large family of highly conserved genes, sharing an elevated degree of sequence homology. The characteristic feature of MAGE proteins is a C-terminal domain present in all the members of the family, termed the MAGE homology domain (MHD). Based on their expression pattern MAGE genes are classified in MAGE-I and MAGE-II genes. MAGE-I genes expression is restricted to tumor and male germ cells, and for this reason they form part of a growing group of genes named Cancer Testis Antigens (CTA). Expression of MAGE-I genes seems to be an early event during gametogenesis and tumorigenesis, and correlates with genomewide hypomethylation, an important event frequently observed in carcinogenesis. Since their discovery in 1991, MAGE-I genes were mostly studied for their potential use in immunotherapy against cancer or as prognostic markers in tumors. The biological roles that these proteins play in tumor development and progression were poorly investigated. Moreover, due to their sequence homology, MAGE-I proteins are still considered functionally redundant proteins. In the present work, we functionally characterized different MAGE-I genes, in particular MageA2 and MageB2 genes, demonstrating their functional specificity. We show that MageA2 protein confers wild-type p53 tumor suppressor-sensitive resistance to chemotherapeutic drugs, such as etoposide, by recruitment of HDAC3 to p53/MageA2 complex, thus repressing p53 transactivation function. The mechanism responsible for the repressive effect of MageA2, relies on an impaired acetylation of both p53 and histones surrounding p53 binding sites by MageA2/HDAC3 complexes. The correlation between MAGE-A expression and resistance to apoptosis has been analyzed in short-term melanoma cell lines, where combined treatment with etoposide and trichostatin A (an inhibitor of histone deacetylases) restores the p53 response and reverts chemoresistance in cells expressing high levels of MAGE-A. We also present evidence that MageA2 is able to repress PML3-induced p53 activity in a specific manner, by affecting PML3 mediated p53 acetylation at the PML3 nuclear bodies (PML3-NBs). The relevance of MageA2 expression on PML3 activity has been analyzed in a normal cellular context, in which PML3 induces premature senescence, an important barrier against cell transformation. In this regard, we demonstrate that MageA2 impairs the senescence response associated to PML3 expression in normal human fibroblast. A possible mechanism for the inhibitory effect of MageA2 on PML3 is that MageA2 could interfere with PML3 sumoylation. The specificity of MageA2 functions is demonstrated by the fact that, despite high level of homology, MageA4 is not recruited to the NBs, it does not affect p53 activity nor is able to interfere with PML3 induced senescence. Finally, we have preliminarily characterized the MageB2 protein, showing that it specifically localizes to the nucleolus where it is able to interact with many nucleolar proteins. Nucleolar stress induces MageB2 relocalization to the nucleoplasm, a characteristic behavior of nucleolar proteins that regulate processes such as rRNA metabolism or RNA processing. Moreover, since we observed that MageB2 induces pRb relocalization to the nucleoli and increases E2F1 transactivation function, including E2F1-induced rRNA transcription, we hypothesize that it could play a positive role in the regulation of cell proliferation. Altogether the work presented here consistently supports the notion that, despite the high level of sequence homology, there is a clear degree of functional specificity within members of the MAGE-I family. Hence, we can hypothesize that different MAGE-I proteins, for instance MageA2 and MageB2, could act within different pathways in the regulation of complex processes such as apoptosis, proliferation, and senescence. By targeting different signal transduction pathways their final outcome could be related to the establishment and progression of the tumors where they are expressed. In this Thesis, we give a comprehensive view on the functional differences among MAGE-I members, focusing on Mage-A and Mage-B members. Implementation of our investigation could be the first step leading to understanding of how expression of specific MAGE-I members could impact cancer cell behaviour thus prompting the use of MAGE-I genes as novel cancer specific targets for the development of new drug-based therapies