I am using esri arc to generate random points. I then analyze the pattern from this process using Average Nearest Neighbor which is also in esri gis but lets say it can be in any other software.
Is there a chance that it comes as dispersed or clustered or will the random generator make sure it is always random?
That's a false distinction.
A random variable generator -- presumably in this case based on independent and uniform probabilities for point coordinates within a defined region -- will produce clustered or dispersed patterns with approximately the probabilities that they are defined to have.
An analogue is generating random integers that are uniform on 1 to 6, as when dice are thrown. All ones or all sixes are expected to occur with their associated (small) probabilities. The larger the number of throws, the smaller the probability of all ones or all sixes, or of any other "extreme" result.
There is no feedback whereby a software function rejects its own results because they have low probabilities in certain senses. That wouldn't even be a coherent idea. Much of the point here is that under a null hypothesis of random patterns (a Poisson point process) you could get patterns that you want to call dispersed or clustered; and you are assessing how likely that is.
So random point patterns necessarily will produce dispersed or clustered point patterns some of the time; the point is rather that the variability is pre-defined so that you can assess that possibility.
Consider also that point patterns do not fall into three distinct classes except for qualitative discussion. There is a continuum from clustered to dispersed.
