How to show special respect to a famous PhD advisor?

he keeps blushing

Simple answer. You are embarrassing him. No-one wants to be embarrassed. Stop doing it and behave normally.

One tangible thing that could do is to nominate your supervisor/advisor for an award if you feel they have done an exemplary job. In my field there are awards for mentorship, research, etc. Sometimes these require a reference letter, which is a significant amount of work, and your supervisor would surely appreciate that.

Edit:

I agree with the other answers cautioning against prematurely elevating a supervisor before you actually being working with them. Like @lighthouse keeper's suggestion with regard to Festschrift, just keep in mind the possibility of award nomination as you progress.

Congratulations! It sounds like your work has already impressed your new supervisor. And after so many years of working for top grades in your undergrad and Master’s, you naturally want to know how to do the same in your PhD.

Here’s the thing: you can’t, not exactly. In the world of research there is usually no unambiguous "top of the class", no one list of achievements that all the best students are expected to aim for.

As a PhD student you will have a project, and over time you’ll get feedback from your supervisor on how you’re doing. From your description, a committee of experts have already expressed their confidence that completing the degree is well within your abilities. There is no scale to measure "how well" you complete it.

After two previous degrees it's a confusing transition, and it's common for students to struggle with this throughout their PhD. I’m sure that astronat is spot on when she invites you to search for information on impostor syndrome, and to take a break. I would also say you’re already going in exactly the right direction by setting achievable goals for yourself: you chose for yourself the target of working with your PhD supervisor, and you got there!

If you want to set yourself some extra targets now, no-one can stop you. But it really sounds like there is nothing more you need to do except recover from all your effort so far. When there is, someone ought to tell you (and not in code, but in plain language).

For completeness, one might mention that there exists a type of publication venue explicitly designed to have contributions that honor an influential researcher: Festschrift. Festschriften are usually bound to a certain event like the celebration a distinguished birthday, like 50 or 60, or the decease of the researcher (in case of which one uses the term ''Gedenkschrift'' instead).

So, you might stash your idea, keep in contact with your professor's other students, and later dig out the idea when the time for it has come.

This is very dependent on your cultural environment (country) and field of studies. In my field and part of the world (Geology, Western cultures) the level of visible marks of respect between student and advisor is very limited - of course, this probably has to do with the fact that during your first field season, you're likely to see your advisor dripping wet half naked out of the shower, covered in mud at the end of a field day or pissed after a beer evening...

This does not mean there is no scientific or professional respect, of course - there is a lot of it- it is not just expressed in very formal ways, and it is routine for geology students to be on first name basis with their supervisor, nearly immediately. And there is certainly a tendency for outside marks or respect to actually decrease with supervisor's reputation (the best do not need to be "Sir-ed" to know that they are respected, those who need the formal respect are more likely than not unsure about their own status).

Now I'm bringing this up as a somewhat extreme example to demonstrate that things can, and do widely differ between communities and countries, so my answer is probably useless to you! :-)

Surely, as Rob Mueller pointed out, the best way is to do good work and good science. At the end of the day (or the career), a top academics will be judged as much by his own achievements than the quality of his students, and the day I'll retire, I'd love to be able to say "yup, I was the PhD supervisor of so-and-so" ....

Best way to show your gratitude and respect would be to do good/great work during this program. Show that he made the right choice.

Even though you have not indicated where your Ph.D. studies are going to be, it is clear from your description of your advisor's behavior that he is not expecting special deference to him.

I would say that the way to show him respect is to trust him when he says "don't do that." Then learn as much as you can in the program, and accomplish great things in your career. That is what he ultimately wants from you.

I have worked with supervisors who are very humble while successful to those who are extremely egoistic due to their accomplishments. In my personal opinion, it is not wise to give Godly status to any person. He is your PhD supervisor, a famous one, but there are thousands of famous people in this world. And fame is subjective. Be professional, be polite and show respect. Do not overdo it as it might cause following issues.

Your colleagues will show distrust in you if you eulogized him, especially if they have a poor relationship with him. By keeping it completely professional you will less likely belong to a specific group in your lab. Same is with other potential supervisors and co-supervisors.

The 3 year PhD journey is not small. There will be ups and downs and without a doubt, there will be times when you will feel poorly treated or exploited. Those time will feel more awful if it comes from the same person who you eulogized.

Most importantly, you must keep your research above person. Write a paper because it needs to be written. Because you want to convey your findings to the community. Not because you wish to impress your supervisor or prove to yourself that you are worthy of doing research. You would end up doing a poor job in your research with that mindset.

You supervisor is just an ordinary human being who did some good research work. There are millions like him. And in a few years, you could be one too.

First of all, Well done and Good luck mate! I hope you make full use of this opportunity.

Secondly, don't be that cautious or THAT MUCH courteous. As you have said that your professor does not care much about what others think of him. Actually, most of the high-profile academics (at least in my humble opinion) think that way. This humbleness, coupled with excessive hard work, take them to the high levels.

They are usually highly professional as well. This is evident in your case as he has himself told you not to do any work because you are not bound to him. It is a give-and-take world mate! You will give him good publications and he will give you good supervision. The rest is being HUMAN - i.e. excelling at everything, be it one's profession or be it one's behaviour. If he is excellent in what he does, plus in his conduct and actions, it shows he is a nice human being. What else can you do to a nice human being than return him the respect, and be nice to other humans imitating him!

The few ways you can give him respect is to be humble yourself, be respectful to him, try to do as much hard work as you can, and later guide the junior researchers that join his lab. Maybe write a few research grants in the third year of your Ph.D. for him. Every now and then giving a few gifts, like a souvenir when you go to any international conferences will also be good.

Disclaimer: Customs and individual preferences can differ, and this is my best guess.

I wouldn't recommend that you show particular deference to your advisor, just because he is famous. Be respectful and polite of course, and deferential to some extent, but it's not necessary to be more so than if your advisor was lesser-known.

I wouldn't recommend reminding him of his own academic status. He is well aware of it already. He will be constantly reminded of this, by the large number of people asking him for recommendation letters, for advice, for input on this or that, wanting to show off their papers to him, etc.

When "he said a few times that he does not need me to do that right now", my best guess is that you should take him at his word. If you have time and energy to spare, you might do some background reading related to the topic of your thesis; this might help you get off to a strong start. Alternatively, you say you are exhausted, so a break might do you a world of good.

Congratulations and good luck!

I don't know what field you're in, but in mine, writing such a paper would be viewed as unnecessary and perhaps a bit strange. You seem to feel that you don't deserve your place with this supervisor, and so you need to prove your worth, or impress them, by writing a paper. This is unecessary, as you have already impressed them with your project idea and PhD application. They would not have taken you on as a student otherwise. Along these lines, you might want to search the term "impostor syndrome", both on this site and on Google.

"I am exhausted...". The last thing you need to be doing right now is writing an unnecessary paper with no help from anyone. A PhD is a several-year-long marathon, and you will need all your energy and stamina to complete it. Take a break, rest and recharge before you start. This will have a far greater benefit to your PhD, and accordingly to your relationship with your supervisor, than writing a paper.

Once you have started the PhD, show your current paper draft to your supervisor and work on it together. The paper will be much better for it. Again, this is field dependent, but in mine it would be extremely rare for a beginning PhD student to be capable of writing a publishable paper entirely on their own. You may be underestimating the challenge. This is something that your supervisor will help you with.

But for now, have a holiday, and celebrate your success at being accepted for a PhD with a great supervisor!