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I hope this question is not too basic and I am not missing something dumb. But suppose we simulated a Turing machine on a long string $s$, where $|s| = 10^{100}$ for example. Then if we wanted to learn the value of $s_i$, the $i$th value in the string, could we do this in say time polynomial in the length of the string?

The issue I am having is differentiating between the theoretical construction of the Turing machine vs. real computers which can for example index arrays in constant time due to their structure in memory. Could a TM obtain $s_i$, the ith value in the array, in time polynomial in $|s|$, regardless of the chosen value of $i$? Or would the head have to "slide over to $i$" with some cap on it's speed, so it could not do this task efficiently?

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This isn't a research-level question, so it's probably better on the normal Computer Science Stack Exchange.

But, to answer the thrust of your question: no, they cannot jump to the middle, and it can be viewed as a limitation of the model. You can use this fact to prove $\Omega(n^2)$ lower bounds on the language of palindromes for a single-tape TM, for example. To avoid this, there are various random access Turing machine models used in the literature.

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