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I found the following definition of equivalence to leading exponential order in the book

Mezard, Marc, and Andrea Montanari. Information, physics, and computation. Oxford University Press, 2009.

The definition is as follows:

The notation $A_N \dot{=} B_N$ will be used throughout the book to denote that two quantities $A_N$ and $B_N$ (which behave exponentially in $N$) are equal to leading exponential order, meaning $\lim_{N\rightarrow\infty} (1/N)\log(A_N/B_N) = 0$.

This is the first time I encounter this definition. Is it in common use somewhere else? Where can I study more about its properties?

a06e
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    That's a fairly standard term, it is a kind of weak asymptotic equivalence. There isn't really much to be said: $A_N$ and $B_N$ are $e^{N f(N)}$ and $e^{N g(N)}$ respectively, and $f(N)$ and $g(N)$ are asymptotically equivalent in the usual sense (i.e. with ratio tending to 1). – Ian Oct 22 '18 at 15:20
  • @Ian Thanks. Can you point to any additional references where this term is used? Ideally textbooks. – a06e Oct 22 '18 at 15:34
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    You can look into any source on large deviation theory; the same concept will be used but it may not be given the same name. I learned a lot from Random Perturbations of Dynamical Systems though this may have a different focus than you really want. – Ian Oct 22 '18 at 15:35
  • @Ian I am reading about large deviation theory and I'm loving it. Very powerful methods. Sometimes you can't find the appropriate keyword to find literature on a topic unless it is pointed out. Thanks a lot. – a06e Oct 29 '18 at 19:50

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