You should move at random and try avoid deterministic strategies, since there's a probability that your friend chooses to move in the exact opposite way and you'll never meet.
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And how do you know that your friend stay and you move ?
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Move at random. Because taking any other of the deterministic options would mean that there’s an inmense probability that your partner took a different one, so you’re never gonna meet. Think of it as the latitude and altitude lines in the sphere. Now, moving randomly means that:
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Whatever option your friend took, you will be more likely to meet him, because you’re describing more possible points for him to meet, think of it as two lines intersecting having much more points in a uniform random distribution (sphere)gives more probability to form thatline.
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By circumnavigating you have two cross points
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If they are travelling a same speed, there are possible scenarios that they'll never meet
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Start digging.
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Follow predefined protocol
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It says you are left with a friend, so you are already in contact with your friend. That is what I call a loophole.
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where communication is impossible. What is the fastest way to meet?
You can justify in the comments