Experimental evidence suggests that nanoscale electronic inhomogeneity plays an important role in a large classes of materials. Local tunneling data indicate strong nanoscale inhomogeneity of superconducting gap in high temperature superconductors. Strong local nanoscale inhomogeneity in the bosonic scattering mode has also been observed in the same samples [1,2]. We argue that these two inhomogeneities are directly related to each other in the overdoped samples. In order to address local boson scattering effects, we would need to develop a local strong coupling model of pairing in a coarse-grained superconducting state. We used Local Inelastic Tunneling Spectroscoy and Fourier Transform Inelastic Tunneling Spectroscopy to address the role of bosonic modes. One implication of this locality is that O¹⁶ --> O¹⁸ isotope substitution effect needs to be considered within this local approximation. We will present a simple model that yields features that are broadly consistent with the doping and isotope substitution trends observed experimentally. We also will discuss isotope effect in a striped phase [3]. To illustrate universality of these questions, we will also discuss possible extensions of similar ideas to observed electronic defects and inhomogeneities in graphene that also exhibits Dirac excitation spectrum [4].

[1] J.-X. Zhu et al., Phys. Rev. B73, 014511 (2006); A. V. Balatsky and J.-X. Zhu, Phys. Rev. B74, 094517 (2006).

[2] J. Lee et al, Nature 442, p 546, (2006).

[3] A. Rosengren, et al, "Isotope effect in a striped superconductor", arXive condmat:0709.4199