Diffusion with stochastic resetting on a lattice

We provide an exact formula for the mean first-passage time (MFPT) to a target at the origin for a single particle diffusing on a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>d</a:mi></a:math>-dimensional hypercubic starting from a fixed initial position <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msub><b:mover accent="true"><b:mi>R</b:mi><b:mo>⃗</b:mo></b:mover><b:mn>0</b:mn></b:msub></b:math> and resetting to <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mover accent="true"><d:mi>R</d:mi><d:mo>⃗</d:mo></d:mover><d:mn>0</d:mn></d:msub></d:math> with a rate <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>r</f:mi></f:math>. Previously known results in the continuous space are recovered in the scaling limit <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>r</g:mi><g:mo>→</g:mo><g:mn>0</g:mn></g:mrow><g:mo>,</g:mo><g:mo> </g:mo><g:mrow><g:msub><g:mi>R</g:mi><g:mn>0</g:mn></g:msub><g:mo>=</g:mo><g:mrow><g:mo>|</g:mo><g:msub><g:mover accent="true"><g:mi>R</g:mi><g:mo>⃗</g:mo></g:mover><g:mn>0</g:mn></g:msub><g:mo>|</g:mo></g:mrow><g:mo>→</g:mo><g:mi>∞</g:mi></g:mrow></g:math> with the product <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:msqrt><i:mi>r</i:mi></i:msqrt><i:mspace width="0.16em"/><i:msub><i:mi>R</i:mi><i:mn>0</i:mn></i:msub></i:mrow></i:math> fixed. However, our formula is valid for any <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mi>r</k:mi></k:math> and any <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:msub><l:mover accent="true"><l:mi>R</l:mi><l:mo>⃗</l:mo></l:mover><l:mn>0</l:mn></l:msub></l:math> that enables us to explore a much wider region of the parameter space that are inaccessible in the continuum limit. For example, we have shown that the MFPT, as a function of <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mi>r</n:mi></n:math> for fixed <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:msub><o:mover accent="true"><o:mi>R</o:mi><o:mo>⃗</o:mo></o:mover><o:mn>0</o:mn></o:msub></o:math>, diverges in the two opposite limits <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mrow><q:mi>r</q:mi><q:mo>→</q:mo><q:mn>0</q:mn></q:mrow></q:math> and <r:math xmlns:r="http://www.w3.org/1998/Math/MathML"><r:mrow><r:mi>r</r:mi><r:mo>→</r:mo><r:mi>∞</r:mi></r:mrow></r:math> with a unique minimum in between, provided the starting point is not a nearest neighbor of the target. In this case, the MFPT diverges as a power law <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"><s:mrow><s:mo>∼</s:mo><s:msup><s:mi>r</s:mi><s:mi>ϕ</s:mi></s:msup></s:mrow></s:math> as <t:math xmlns:t="http://www.w3.org/1998/Math/MathML"><t:mrow><t:mi>r</t:mi><t:mo>→</t:mo><t:mi>∞</t:mi></t:mrow></t:math>, but very interestingly with an exponent <u:math xmlns:u="http://www.w3.org/1998/Math/MathML"><u:mrow><u:mi>ϕ</u:mi><u:mo>=</u:mo><u:mo>(</u:mo><u:mo>|</u:mo><u:msub><u:mi>m</u:mi><u:mn>1</u:mn></u:msub><u:mo>|</u:mo><u:mo>+</u:mo><u:mo>|</u:mo><u:msub><u:mi>m</u:mi><u:mn>2</u:mn></u:msub><u:mo>|</u:mo><u:mo>+</u:mo><u:mo>...</u:mo><u:mo>+</u:mo><u:mo>|</u:mo><u:msub><u:mi>m</u:mi><u:mi>d</u:mi></u:msub><u:mo>|</u:mo><u:mo>)</u:mo><u:mo>−</u:mo><u:mn>1</u:mn></u:mrow></u:math> that depends on the starting point <v:math xmlns:v="http://www.w3.org/1998/Math/MathML"><v:mrow><v:msub><v:mover accent="true"><v:mi>R</v:mi><v:mo>⃗</v:mo></v:mover><v:mn>0</v:mn></v:msub><v:mo>=</v:mo><v:mi>a</v:mi><v:mspace width="0.16em"/><v:mrow><v:mo>(</v:mo><v:msub><v:mi>m</v:mi><v:mn>1</v:mn></v:msub><v:mo>,</v:mo><v:msub><v:mi>m</v:mi><v:mn>2</v:mn></v:msub><v:mo>,</v:mo><v:mo>...</v:mo><v:mo>,</v:mo><v:msub><v:mi>m</v:mi><v:mi>d</v:mi></v:msub><v:mo>)</v:mo></v:mrow></v:mrow></v:math> where <y:math xmlns:y="http://www.w3.org/1998/Math/MathML"><y:mi>a</y:mi></y:math> is the lattice spacing and <z:math xmlns:z="http://www.w3.org/1998/Math/MathML"><z:msub><z:mi>m</z:mi><z:mi>i</z:mi></z:msub></z:math>'s are integers. If, on the other hand, the starting point happens to be a nearest neighbor of the target, then the MFPT decreases monotonically with increasing <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML"><ab:mi>r</ab:mi></ab:math>, approaching a universal limiting value 1 as <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML"><bb:mrow><bb:mi>r</bb:mi><bb:mo>→</bb:mo><bb:mi>∞</bb:mi></bb:mrow></bb:math>, indicating that the optimal resetting rate in this case is infinity. We provide a simple physical reason and a simple Markov-chain explanation behind this somewhat unexpected universal result. These interesting results on a lattice are not captured by the continuum theory. Our analytical predictions are verified in numerical simulations on lattices up to 50 dimensions. Finally, in the absence of a target, we also compute exactly the position distribution of the walker in the nonequlibrium stationary state that also displays interesting lattice effects not captured by the continuum theory.