What is vacuum to DC flow?

I know how an air gap behaves: for lower voltages the resistance is extremely high, until spark voltage is achieved, at which point resistance rapidly drops, creating electric arc. Now, how does that behave if we remove air?

Let's assume a part of electronic circuit that is a vacuum pipe with two electrodes at its ends. The electrodes are otherwise separated by an extremely (infinitely) highly resistive material. What would the I:U graph of voltage across the void gap versus amperage of current flow look like for element like this? What would it depend on, besides voltage (and possibly distance between the electrodes)?

As I understand some charge flows through vacuum between oppositely charged electrodes but it's nowhere near as straightforward as in case of normal conductors. How does it look like? What are the caveats? Is there a plain equivalent in common circuitry that behaves just the same as a void gap for common electrical characteristics (voltage-resistance curve etc) ?

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Does a vacuum tube still count as "common circuitry"? Because, you know... – Ilmari Karonen Sep 16 '13 at 11:17
@IlmariKaronen: More or less yes, but with currents of heating the cathode varying between "nominal" and "zero" for answering my question more completely. In particular, what are the characteristics with cold cathode? – SF. Sep 16 '13 at 11:27