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I need some good books aimed as a detailed and gentle introduction to random matrices, containing good discussion and derivation of Marchenko–Pastur distribution. Also, I request some other references on the application side that discuss the applications of random matrices to fields like finance, random polynomials and physics. I would be highly obliged for any recommendations .

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If you are interested in doing research in this field, you will want to learn of major applications, and for that purpose I could recommend the various chapters of The Oxford Handbook of Random Matrix Theory. The book is expensive, but many of the chapters are on arXiv. In particular, part III includes a great number of applications.

Here is a list of 17 chapters from the Handbook on arXiv, including chapters on financial applications and on physics applications.

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