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We present novel statistical tools to cross-correlate frequency cleanedthermal Sunyaev-Zel'dovich (tSZ) maps and tomographic weak lensing (wl)convergence maps. Moving beyond the lowest order cross-correlation, weintroduce a hierarchy of mixed higher-order statistics, the cumulants andcumulant correlators, to analyze non-Gaussianity in real space, as well ascorresponding polyspectra in the harmonic domain. Using these moments, wederive analytical expressions for the joint two-point probability distributionfunction (2PDF) for smoothed tSZ (y_s) and convergence (\kappa_s) maps. Thepresence of tomographic information allows us to study the evolution of higherorder {\em mixed} tSZ-weak lensing statistics with redshift. We express thejoint PDFs p_{\kappa y}(\kappa_s,y_s) in terms of individual one-point PDFs(p_{\kappa}(\kappa_s), p_y(y_s)) and the relevant bias functions(b_{\kappa}(\kappa_s), b_y(y_s)). Analytical results for two different regimesare presented that correspond to the small and large angular smoothing scales.Results are also obtained for corresponding {\em hot spots} in the tSZ andconvergence maps. In addition to results based on hierarchical techniques andperturbative methods, we present results of calculations based on the lognormalapproximation. The analytical expressions derived here are generic andapplicable to cross-correlation studies of arbitrary tracers of large scalestructure including e.g. that of tSZ and soft X-ray background.

Cross-correlating Sunyaev-Zel'dovich and Weak Lensing Maps