Dear Vamsi,

A while ago I wrote my point of view on what "you should and shouldn't read" before studying Arakelov geometry. See

http://mathoverflow.net/questions/54603/what-should-i-read-before-reading-about-arakelov-theory/54615#54615

Taking another look at that answer, it seems that my answer is  written for people with a more algebraic background.  I think the "road to Arakelov geometry" for someone from analysis is a bit different, but I'm convinced that **a** way to start is the same for everyone.

If you're more comfortable with analysis than algebraic geometry, I think  a good idea would be to start with the analytic part of Arakelov geometry. This is explained very well in Chapter 1.1 of R. de Jong's thesis

http://www.math.leidenuniv.nl/~rdejong/publications/

and P. Bruin's master's thesis (written under the supervision of R. de Jong and B. Edixhoven)

http://www.math.leidenuniv.nl/~pbruin/

These two explain very well what Faltings and Arakelov did in their articles. 

Since you don't want to apply the analysis to do intersection theory on an arithmetic surface, you don't have to go into this, I believe. (This is where schemes and number theory comes into play.)

Now, I think after reading the relevant parts in the above references, you could start reading papers about analytic torsion (assuming you're already familiar with these). There's many of these, but I'm not the person to tell you which one is the best to start with.

Good luck!