I can't duplicate your problem on a Uno. I've not used the analog comparator before so it was an interesting experiment for me. I've tried with the test sketch below. If the voltage on AIN0 is less than that of AIN1, then there is no false trigger even if the Arduino is reset.
AIN0 = Pin D6
AIN1 = Pin D7
LED on D12.
If you have actually measured voltages, then it is difficult to imagine a wiring error. Of course, a potentiometer will not be 100% smooth as you are adjusting the value so you will get mixed rising and falling edges.
// as Nick Gammon's sketch but also with a LED
volatile boolean triggered;
ISR (ANALOG_COMP_vect)
{
triggered = true;
}
void setup ()
{
pinMode( 12, OUTPUT ) ; // led on pin 12
Serial.begin (115200);
Serial.println ("Started.");
ADCSRB = 0; // (Disable) ACME: Analog Comparator Multiplexer Enable
ACSR = bit (ACI) // (Clear) Analog Comparator Interrupt Flag
| bit (ACIE) // Analog Comparator Interrupt Enable
| bit (ACIS1); // ACIS1, ACIS0: Analog Comparator Interrupt Mode Select (trigger on falling edge)
} // end of setup
void loop ()
{
digitalWrite( 12, ACSR & bit (ACO) ) ; // 0b00100000
if (triggered)
{
Serial.println ("Triggered!");
triggered = false;
}
} // end of loop