Is there an exact definition or measure of what a talmid chochom is? For example the gemara in berachos (daf 43b) says that there are 6 things that are degrading for a talmid chochom. How is a talmid chochom defined? Also, do we pasken that a talmid chochom should not talk with his wife in the market? If so, what is the definition of talking (ie is it an extended conversation with laughing etc or even an exchange of a few words?) and what is the definition of market? Does it include going to a shopping centre (which would mean that a talmid chochom can basically never go shopping with his wife)
The question of who is a talmid chacham is involved, and varies from issue to issue. For instance, see Shach, Choshen Mishpat 83:41, who writes concening one matter (making an oath in beis din) that a talmid chacham is somebody who wears tefillin. However, in general the definition of a talmid chacham is somebody who is able to rule halachos, is G-d fearing, and is worthy of teaching the multitudes (Rivash 1:147). Or Le-Zion (Vol. 2, 8:1, in footnote) adds that it is sufficient that he rules halachos from sefarim.
Concerning speaking to women in the market, a marketplace is a public area. Yet, with regard to today’s talmidei chachamim, the halachah is mentioned in Rambam (Deos), but is difficult to apply today. As Rabbi Chaim Berlin writes (Nishmas Chaim 135:5), the prohition only applies to somebody who is renowned to be a talmid chacham, and for somebody who is not renowned as such, acting in this manner would be a chilul Hashem. It is crucial, he continues to write, for a person to live among men, and in particular one who is known to be righteous must beware of bringing ridicule upon the G-dfearing. An almost identical statement is made by Rav Y. C. Zonnenfeld (Salmas Chaim 706). [See also Az Nidberu 3:65.]
In Hilchos Niddah (YD 195) we are taught that a person should not stoll with his wife while she is prohibited to him, indicating that in general there is no prohibition. Indeed, there is a known anecdote of the Beis Yisroel and the Nesivos Shalom: the latter used to stroll with his wife, and the Beis Yisroel commented on the practice, yet the Nesivos Shalom was clearly not concerned. Today, it is common practice for talmidei chachamim to stroll (and even to go shopping) with their wives, and it is well-known that today the practice is not considered shameful.