What is the purpose of using a 1M Ohm resistor across crystal pins? Please refer to the image for clarity. The crystal serves as the clock source for an USB Hub device(LAN9512). There is nothing mentioned in the HUB datasheet about this, I suppose they have added it based on their experience, I have no clue about it.
I found this document (Schematic Checklist), from the chip manufacturer (Microchip) which states (note point 5).
Usually something like a high-value bias resistor is supplied internally for a Pierce oscillator configuration typically used on IC crystal oscillators, but some ICs don't have them, and perhaps this chip either does not have one, or the ones on that chip revision were considered inadequate for some reason. That's speculation on my part, but the bottom line is that if Microchip says to use it, you'd better use it.
The bias resistor (internal or external) is required for the oscillator to reliably start up- it biases the amplifier into the linear region where noise can be amplified to get the crystal oscillator going.
IF an IC manufacturer specifies that a resistor is needed then it probably is. If "only" a cct designer says so then it may or may not be so. If they have gone to the effort of very specifically stating the need it may be a real one. As Spehro said, biasing of the internal electronics may be required - the OC manufacturer should specify this if this is the case. Datasheets for ICs in eg USB hubs may be 'hard to find' In such cases a manufacturer's experience is liable to count (often, not always).
Some crystals are unhappy with "high drive power" circuits and some crystals are "high activity". It is possible to physically damage some crystals in some circuits from the mechanical energy involved. In such cases, adding a damping resistor can reduce cct Q and reduce the chance of crystal damage. This is usually a problem with olde world steam driven / valve circuits and less common in modern IC based circuits.
I was also looking into this myself and found some more info that could help. From here, https://rheingoldheavy.com/arduino-from-scratch-part-9-16mhz-crystal-oscillator/, similar to @Spehro's answer above,
This resistor is actually acting as a feedback resistor for the micro’s internal inverter to which the crystal is connected, and biases the inverter’s input into the linear region. “Biasing into the linear region”, means that it is amplifying the crystal’s oscillation without ever fully saturating at one of the voltage rails. When power is first applied to the system, this feedback resistor helps the components that make up the oscillator circuit, both internal to the micro and the external parts we provide, to get ringing at the correct frequency faster.
Also, this PDF (http://www.crystek.com/documents/appnotes/Pierce-GateIntroduction.pdf) shows how you could design for a value of resistance for a given frequency; that document's table should at least give a ballpark of what can be considered reasonable.